This page is the beginnings of an

Encyclopedia of Combinatorial Polytope Sequences...
...but it is growing. Send entry suggestions to: Entry Editor

Searching the table:
  • Ctrl-F, with sub-sequence using commas and spaces: 0, 3, 6
  • or use the index of named sequences
  • or enter below any keyword or sub-sequence: vertices, facets, polygon faces or f-vectors. Use quotes: "0, 3, 6, 0" to find named entry. 
    








 

Each entry of this encyclopedia is a well defined sequence P_n , array or otherwise indexed family of polytopes with a combinatorial definition.  Duals (polars) to the polytopes are considered to be the same entry--just reverse the f-vectors.  



Pictured are the 3d terms of polytope sequences, which have at most one term in each dimension. Pictures are often links to individual pages.  


Below each picture is a list giving the polygon score in the 3d term: (# triangles, # squares, # pentagons, # hexagons, ...)


Immediately to the right is a list of names for the sequence: many sequences have multiple interpretations.


The sequences of numbers of vertices and facets in each dimension n begins with n=0.


The sequence labeled f-vectors is the triangle of f-vectors read by rows: each row starts with vertices, and the occurrences of 1 are the top-dimension faces.


Links are to introductory literature, not necessarily primary sources.



 
SEQUENCES

 
Simplex






	  

(4, 0, 0, 0)

 



  •   Simplices Δ [wiki]

    

  •   Multi-associahedron for n=2k+2 [arxiv] (V. Pilaud)
    

  •   Order polytope O(P) for P the linear order on {1,...,n}. [citeseer](R.Stanley)

    

  •   Chain polytope C(P) for P the linear order on {1,...,n}. [citeseer](R.Stanley)

    

  •  Poset associahedra for antichain [arxiv]
    

  •  Vertex cover polytope of the complete graph VC(K_n) [wiki]
    

  •  edgeless-graph-associahedra [arxiv] (S. Devadoss)

    

  •  (n+1)-cycle-graph graphic matroid polytopes [wiki] 
    

  •  Uniform matroid U^n_(n+1) polytope [wiki] 

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 3, 4, 5, ... n+1 [ OEIS A000027 ]

    

  •  Numbers of facets in dimension n (starting at n=0):
      0, 2, 3, 4, 5, ... n+1 [ OEIS A000027 ]

    

  •  f-vectors:
     1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, ... [ OEIS A135278]

    		• 

    

    

    top     index 
     

    

Demihypercube

    


    

	  
    
(4, 0, 0, 0)
    		
 

    

  •   Demihypercubes; [wiki]

    

  •  n-demicubes, n-hemicubes  
    

  •  convex_hull({alternating vertices of n-cube}) 
    
 

    		• 

  •  Dimensions: 
     1, 3, 4, ... n
    

  •  Number of Vertices in nth polytope:
     2, 4, 8, 16 ... 2^n [ OEIS A000079]

    

  •  Numbers of facets in dimension n
    
  2, 4, 16, 26, 44, 78,... 2^(n-1)+2n [ OEIS ?]

    

  •  f-vectors:
     1, 2, 1, 4, 6, 4, 1, 8, 24, 32, 16, 1, 16, 80,... [ OEIS ?]

    		• 

    

    

    top     index 
     

    


Cut Polytope

    


    

	  
    
(4, 0, 0, 0)
    
[polymake for n=4]
    		
 

    

  •  Cut Polytopes of complete graph  CUT(n)= P_C(K_n)  [Springer] (Barahona, Mahjoub)
	[SMAPO library]
	

    

  •  Correlation Polytopes COR(n)
     

  •  convex_hull({incidence_vector_F | F a cut of the complete graph on n nodes})
    

    		• 


  •  Dimensions: 
     1, 3, 6, 10, 15,... (n choose 2) [ OEIS A000217]
    

  •  Number of Vertices in nth polytope:
    2, 4, 8, 16, 32 ... 2^(n-1) [ OEIS A000079]

    

  •  Number of Facets:
     2, 4, 16, 56, 368, 116764, ... OPEN [ OEIS A235459]
    

  •  f-vectors: 
     1, 2, 1, 4, 6, 4, 1, 8, 28, 56, 68, 48, 16, 1... [ OEIS ?]

    		• 

    

    

    top     index 
     

    

Flow Polytope

    


    

	  
    
(4, 0, 0, 0)
    
[polymake for n=5,6]
    		
 

    

  •  Flow Polytopes of complete graph FLOW(n)= F(K_n)  [arxiv] (Mészáros,  Morales, Striker)
    

  •  Chan-Robbins-Yuen Polytopes CRY(n-1)
     

  •  convex_hull({incidence_vector_F | F a unit flow of the complete graph on n nodes})
    

  •  Volume equals product of the first n - 2 Catalan numbers

    		• 


  •  Dimensions: 
     1, 3, 6, 10, 15,... (n-1 choose 2) [ OEIS A000217]
    

  •  Number of Vertices in nth polytope:
    2, 4, 8, 16, 32 ... 2^(n-2) [ OEIS A000079]

    

  •  Number of Facets:
     2, 4, 8, 13 ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 4, 6, 4, 1, 8, 26, 45, 45, 26, 8, 1, 16, 98, 327, 681, 944, 897, 588, 262, 76, 13... [ OEIS ?]

    		• 

    

    

    top     index 
     

    

Matching Polytope of complete graph

    


    

	  
    
(4, 0, 0, 0)
    
[polymake for n=4]
    		
 

    

  •  Matching Polytopes of complete graph MATCH(n)= M(K_n)  [wiki] 
    

  •  convex_hull({incidence_vector_M | M a general matching of the complete graph on n nodes}) [imsc](M. Mahajan)
    

    		• 


  •  Dimensions: 
    0, 1, 3, 6, 10, 15,... (n choose 2) [ OEIS A000217]
    

  •  Number of Vertices in nth polytope:
    1, 2, 4, 10, 26, 76, 232, 764... Sum_{k=0..[ n/2 ]} n!/((n-2*k)!*2^k*k!) [ OEIS A000085]

    

  •  Number of Facets:
     2, 4, 14... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 4, 6, 4, 1, 10, 39, 78, 86, 51, 14,... [ OEIS ?]

    		• 

    

    

    top     index 
     

    



Hypercube

    


    

	  
    
(0, 6, 0, 0)
    		
 

    

  •  Cubes  C   [wiki]
    
	

  •   Order polytope O(P) for P the poset with no relations on n elements (antichain with n elements). [citeseer](R.Stanley)

    

  •   Cayley polytopes C_j [arxiv](Beck, Braun, Le)
    

  •   Chain polytope C(P) for P the poset with no relations on n elements (antichain with n elements). [citeseer](R.Stanley)
    

  •  Lipschitz polytope L(P) for P an antichain.[Sanyal and Stump]
    

  •  Lipschitz polytope L(P) for P a chain.
    

  •  Stanley-Pitman polytopes Pi_n(x) [arxiv](Pitman, Stanley), [arxiv] (Postnikov, Reiner, Williams)
    

  •  Acyclotopes, or graphical zonotopes, for graphs that are forests. [ Zaslavsky ], [ citeseer ], [ Postnikov ]
     

  •  Voronoi cells of cographical lattice for tree graphs (primary parallelohedra, primary parallelotopes) [F. Vallentin]
    

  •   Brillouin zone (Wigner-Seitz cell of reciprocal space) for Simple Cubic lattice in 3d [wiki]

    

  •  Poset associahedra for cross-stack posets [arxiv]
    

  •  Quotientopes  P , whose upper ideal of shards  contains only the basic shards.
[Pilaud, Santos]
    

  •  (0,n)-Hochschild polytopes Hoch(0,n) [ arxiv ] Pilaud, Poliakova	

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
     

  •  Number of Vertices in nth polytope:
     1, 2, 4, 8, 16, ... 2^n [ OEIS A000079 ]

    

  •  Number of Facets (start at n=0): 
     0, 2, 4, 6, 8 ... 2*n  [ OEIS A004277]
    

  •  f-vectors:
     1, 2, 1, 4, 4, 1, 8, 12, 6, 1, 16, ... [ OEIS A038207]

    		• 

    

    

    top     index 
     

    

Independent set polytope of Uniform matroid

    


    

	  
    
(4, 3, 0, 0)
    		
 

    

  •  Uniform matroid U^(n-1)_n independent set polytope [arxiv] (Ardila, Benedetti, Doker)

    

  •  n-cycle-graph graphic matroid independent set polytopes [wiki] 

    		• 

  •  Dimensions: 
     1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 3, 7, 15, ... 2^n - 1 [ OEIS A000225]

    

  •  Number of Facets (start at n=0): 
     0, 0, 3, 7, ...

    		• 

    

    

    top     index 
     

    




Bipartite Subgraph Polytope 

    


    

	  
    
(4, 3, 0, 0)
    		
 

    

  •  Bipartite Subgraph Polytopes of the complete graph P_B(K_n) =  BS(n)
    [jstor](F. Barahona, M. Grötschel, A. Mahjoub)
	[SMAPO library(large subgraphs only)]

    

    		• 


  •  Dimensions: 
     0, 1, 3, 6, 10 ... (n choose 2)
    

  •  Number of Vertices in nth polytope:
     1, 2, 7, 41, 376, ... [ OEIS A047864]

    

  •  Number of Facets (start at n=1): 
     0, 2, 7, ...

    		• 

    

    

    top     index 
     

    
	
Tutte Polytope

    


    

	  
    
(2, 5, 0, 0)
    
[polymake for 3D]
    		
 

    

  •  Tutte Polytopes T_i [arxiv]
	(Konvalinka, Pak)
    


  •  Volumes equal evaluated Tutte polynomials of complete graphs.

    		• 


  •  Dimensions: 
     1, 2, 3, 4, ... n 
    

  •  Number of Vertices in nth polytope:
    2, 4, 8, 16, 32, ... 2^n [ OEIS A000079]

    

  •  Number of Facets:
     2, 4, 7, 11, 16, 22, 29, ... (n+1)n/2 + 1 [ OEIS A000124]
    

  •  f-vectors: 
     1, 2, 1, 4, 4, 1, 8, 13, 7, 1, 16, 37, 32, 11, 1, ... [ OEIS ?]

    		• 

    

    

    top     index 
     

    	

Cubeahedron (edgeless graph)

    


    

	  
    
(1, 3, 3, 0)
    		
 

    

  •  Edgeless-graph cubeahedra [arxiv] (Devadoss, Heath, Vipismakul)

    

  •  Range quotient of edgeless-graph multiplihedron JGr
[arxiv] (Devadoss, Forcey) 
    		• 


  •  Dimensions: 
     1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     2, 5, 10, ... 2^n + n-1 [ OEIS A052944 ]

    

  •  Number of Facets (start at n=0): 
     0, 2, 5, 7, 9, ... 2*n + 1 [ OEIS A130773 ]

    		• 

    

    

    top     index 
     

    



Linear Ordering Polytope

    


    

	  
    
(8, 0, 0, 0)
    		
 

    

  •  Linear ordering polytopes  P_LO  [zib.de] (M. Grötschel, M. Jünger, G. Reinelt),
     [citeseer] (T. Christof, G. Reinelt),
[arxiv] (Katthän),  [SMAPO library]

    

  •  Binary choice polytopes 
    

  •  convex_hull({char_vector_LO | LO a linear order with n elements}) 
    

    		• 


  •  Dimensions: 
     0, 1, 3, 6, 10, ... (n choose 2)
    

  •  Number of Vertices in nth polytope:
    1, 2, 6, 24, ... n! [ OEIS A000142][see: i, ii, iii] 

    

  •  Number of Facets:
     0, 2, 8, 20, 40, 910, 87472  ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 6, 12, 8, 1, 24, ... [ OEIS ?]

    		• 

    

    

    top     index 
     

    

Acyclotope (Tadpole graph)

    


    

	  
    
 (0, 6, 0, 2)
    		
 

    

  •  Quotientopes  P , whose upper ideal of shards  contains the basic shards,
     and  (1, 3, {2}), and (1, 3, {}). 
[Pilaud, Santos]

    

  •  Acyclotopes  A(T_3,n)  for tadpole graphs T_3,n, with n+3 nodes.
 [ Zaslavsky ], [citeseer]

    

  •  Graphical zonotopes for tadpole graphs Z(T_3,n)
 [Postnikov]

    

  •  Voronoi cells of cographical lattice for tadpole graphs T_3,n (primary parallelohedra, primary parallelotopes) [F. Vallentin]
    

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n+2
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 12, 24, 48, ...  6*2^n  acyclic orientations of the tadpole graph on n+3 nodes[ OEIS A007283]

    

  •  Number of Facets:
     0, 2, 6, 10, 14,  ... 6+2n directed edge cuts of the tadpole graph on n+3 nodes [ OEIS A005843]

    		• 

    

    

    top     index 
     

    







Freehedron 

    


    

	  
    
(0, 4, 4, 0)
    		
 

    

  •  Freehedra  F  [arxiv] (Saneblidze)

    

  •  
  •  (1,n)-Hochschild polytopes Hoch(1,n) [ arxiv ] Pilaud, Poliakova	 
     

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    


  •  Number of Vertices in nth polytope:
    2, 5, 12, 28, 64, 144, 320, 704...(n+3)*2^(n-2) [Conj. OEIS A045623] (F. Chapoton) 

    

  •  Number of Facets:
     0, 2, 5, 8, 11 ... 3*n - 1 [ OEIS A016789] 

    		• 

    

    

    top     index 
     

    





Associahedron 

    


    

	  
    
( 0, 3, 6, 0)
    		
 

    

  •  Associahedra  K, Y  [claymath] (J.L. Loday)

    

  •  Stasheff polytopes
     

  •  Type A associahedra  [arxiv] (Fomin, Reading)

    

  •  Amplituhedron for biadjoint scalar phi^3 theory [arxiv](Arkani-Hamed, Bai, He,Yan)
    

  •  Secondary polytope of the set of vertices of a polygon [maa review]  (Gelfand, Kapranov, Zelevinsky)
    

  •  Newton polytope of the discriminant polynomial of the (integer coordinate) vertices of a polygon.
    

  •  Multi-associahedron Delta(n,1) for k=1. [arxiv] (Pilaud, Santos)
    

  •  Fiber polytope of the simplex over a polygon. [jstor] (Billera, Sturmfels)
    

  •  Path graph associahedra  [arxiv] (Carr, Devadoss)

    

  •  Path graph cubeahedra  [arxiv] (Devadoss, Heath, Vipismakul)

    

  •  Zig-zag poset associahedra  [arxiv] (Devadoss et.al.)

    

  •  Chain (Galashin) poset associahedra [arxiv] (P. Galashin)
	
    

  •  Quotientopes  P , whose upper ideal of shards  contains the basic shards and all upper shards.
[Pilaud, Santos]

    

  •  2-associahedra W_n for the sequence(n). [arxiv](N. Bottman) 

     

  •  (1,n) biassociahedra  KK(n,1), KK(1,n)  [arxiv] (Saneblidze, Umble)

    

  •  alt. notation  B^n_1,  B^1_n  [arxiv] (Markl)

    

  •  Constrainahedra  C(1,n) = C(n,1),
[ arxiv ] ( Bottman, Poliakova)
        or  Constr(0,n) = Constr(n,0), [ arxiv ] (  Chapoton, Pilaud)

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 5, 14, 42, ...  Catalan numbers [ OEIS A000108]

    

  •  Number of Facets:
     0, 2, 5, 9, 14, ...  Triangular numbers minus one  [ OEIS A000096 ]

    

  •  f-vectors: 
     1, 2, 1, 5, 5, 1, 14, 21, 9, 1, 42, 84, 56, 14, 1, ... [ OEIS A033282]
    

  •  h-vectors: 
     1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, 1, 15, 50, 50, 15, 1... [ OEIS A001263]
    

    		• 

    

    

    top     index 
     

    

Type D Associahedron 

    


    

	  
    
( 0, 3, 6, 0)
    		
 

    

  •  Type D associahedra  [arxiv] (Fomin, Reading) [arxiv] (Ceballos, Pilaud)

    		• 

  •  Dimensions: 
     2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
    4, 14, 50, 182 ... (3n-2)*C(n-1),  where C is Catalan numbers [ OEIS A051924]

    

  •  Number of Facets:
     4, 9, 16, 25 ...  n^2   [ OEIS A000290]

    

  •  f-vectors: 
     1, 4, 4, 1, 14, 21, 9, 1, 50, 100, 66, 16, 1, ... [ OEIS A080721]

    		• 

    

    

    top     index 
     

    

Multipath (pseudograph) associahedron

    


    

	  
    
 (0, 5, 2, 2)
    		
 

    

  •  2-path associahedra  P(n,2); P(n,1) , for the multipath formed by doubling all edges
     of the path on n nodes, or respectively all but the last edge.
  [arxiv] Carr, Devadoss, Forcey 

    

  •  2-associahedra  W_1, W_10, W_101, W_1010, ...  .
 [arxiv](N. Bottman) 

    

  •  Poset associahedra for the poset of the 2-paths.
  [arxiv] (Devadoss et.al.)

    

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 4, 14,  ...  ? [ OEIS ?]

    

  •  Number of Facets:
     0, 2, 4, 9,   ...  ? [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 4, 4, 1, 14, 21, 9, 1, ... [ OEIS ?]

    		• 

    

    

    top     index 
     

    


Composihedron 

    


    

	 
    
(0, 4, 6, 0)
    		
 

    

  •  Composihedra  CK   [arxiv] (Forcey)

    

  •  Path-graph composihedra [arxiv] (Devadoss, Forcey)
    

  •  (in low dimensions) pasting diagrams of pseudomonoids in monoidal 2-categories [TAC] (P. McCrudden)
    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 5, 15, 51, ...   binomial transform of Catalan numbers [ OEIS A007317]

    

  •  Number of Facets:
    0, 2, 5, 10, 19 ...  2^n+n-1  [ OEIS A052944]

    		• 

    

    

    top     index 
     

    

Halohedron

    


    

	  
    
 (0, 3, 6, 1)
    		
 

    

  •  Halohedra  H  [arxiv] (Devadoss, Heath, Vipismakul)

    

  •  Cycle-cubeahedra 
[arxiv] (Devadoss, Forcey)
    

  •  1-loop Amplituhedron for planar ϕ^3 theory 
[arxiv] (Salvatori)
	

    		• 


  •  Dimensions: 
     1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     2, 5, 16, 55, 196, ... Catalan(n)*(3n+2)[ OEIS A051960]

    

  •  Number of Facets:
    0, 2, 5, 10, 17, ...  n^2+1  [ OEIS A002522]

    		• 

    

    

    top     index 
     

    


Graph composihedron (cycle graph)

    


    

	  
    
 (0, 3, 6, 1)**
    		
 

    

  •  Cycle-composihedra  JGd  
 [arxiv] (Devadoss, Forcey)

    

    		• 

  •  Dimensions: 
     1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     2, 5, 16, 61 ... OPEN [ OEIS ?]

    

  •  Number of Facets:
     0, 2, 5, 10, ... OPEN [ OEIS ?]

    		• 

    

    

    top     index 
     

    

Stellohedron

    


    

	  
    
 (0, 3, 6, 1)
    		
 

    

  •  Stellohedra  S   [arxiv] (Postnikov, Reiner, Williams)

    

  •  Secondary polytopes of pairs of nested concentric n-dimensional simplices. [arxiv] (V. Pilaud, T. Manneville)
    

  •  Star-graph associahedra [arxiv] (Carr, Devadoss)

    

  •  complete-graph-cubeahedra [arxiv] (Devadoss, Heath, Vipismakul)

    

  •  complete-graph-composihedra JGd  [arxiv] (Devadoss, Forcey)

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 5, 16, 65,  ...  Sum_{k=0..n} n!/k!   [ OEIS A000522][see: i]

    

  •  Number of Facets:
    0, 2, 5, 10, 19, 36, ... 2^n + n - 1 [ OEIS A052944 ] (Thanks to P. Showers)
    

  •  f-vectors: 
     1, 2, 1, 5, 5, 1, 16, 24, 10, 1, 65, 130, ... [ OEIS A248727] (Thanks to Tom Copeland)
    

  •  h-vectors: 
     1, 3, 1, 1, 7, 7, 1, 1, 15, 33, 15, 1, 1, 31, 131, 131, 31, 1, ...[ OEIS A046802] (Thanks to Tom Copeland)
    


    		• 

    

    

    top     index 
     

    


Graph composihedron (edgeless graph)

    


    

	 
    
 (1, 6, 3, 0)
    		
 

    

  •  Edgeless-graph-composihedra  JGd  
 [arxiv] (Devadoss, Forcey)

    

    		• 

  •  Dimensions: 
     1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     2, 5, 13, ... OPEN [ OEIS ?]

    

  •  Number of Facets:
     0, 2, 5, 10, ... OPEN [ OEIS ?]

    		• 

    

    

    top     index 
     

    


Graph multiplihedron (edgeless graph)

    


    

	 
    
 (2, 6, 0, 3)
    		
 

    

  •  Edgeless-graph-multiplihedra  JG  
 [arxiv] (Devadoss, Forcey)

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 6, 15, 36, ...  n*2^(n-1) + n  [ OEIS A215149]

    

  •  Number of Facets:
     0, 2, 6, 11, 20, ...  2^n + n  [ OEIS ?]

    		• 

    

    

    top     index 
     

    


Cyclohedron

    


    

	  
    
 (0, 4, 4, 4)
    		
 

    

  •  Cyclohedra  W  
 [arxiv] (M. Markl)

    

  •  Bott-Taubes polytopes
     

  •  Type B,C associahedra 
 [arxiv] (S. Fomin, N. Reading)(R. Simion)

    

  •  cycle-graph-associahedra [arxiv] (S. Devadoss)
    

  •  (in low dimensions) hexagonator equations (pasting diagrams) in braided monoidal categories [arxiv] (M. Stay)
    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 20, 70, ...  central binomial coefficients  [ OEIS A000984]

    

  •  Number of Facets:
     0, 2, 6, 12, 20,  ...  n^2+n  [ OEIS A002378]
    

  •  f-vectors: 
     1, 2, 1, 6, 6, 1, 12, 30, 20, 1, 20, 90, 140, ... [ OEIS A063007]

    		• 

    

    

    top     index 
     

    

Acyclotope (cycle graph)

    


    

	  
    
 (0, 12, 0, 0)
    		
 

    

  •  Acyclotopes for cycle graph  A(G)  
[ Zaslavsky ],  [ citeseer ]

    

  •  Graphical zonotopes for cycle graph Z(G)
 [Postnikov]

    

  •  Voronoi cells of cographical lattice for cycle graphs (primary parallelohedra, primary parallelotopes) [F. Vallentin]
    

  •   Brillouin zone (Wigner-Seitz cell of reciprocal space) for Body Centered Cubic lattice in 3d [wiki]

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 14, 30, ...  acyclic orientations of (n+1)-cycle = 2^(n+1) - 2  [ OEIS A000918]

    

  •  Number of Facets:
     0, 2, 6, 12, 20,  ... directed edge cuts of the (n+1)-cycle = n^2+n  [ OEIS A002378]

    		• 

    

    

    top     index 
     

    

Acyclotope (zigzag ladder graph)

    


    

	  
    
 (0, 8, 0, 4)
    		
 

    

  •  Quotientopes  P , whose upper ideal of shards  contains the basic shards, and  (i, i+2, {i+1}), and (i, i+2, {}). 
[Pilaud, Santos]

    

  •  Acyclotopes  A(G)  for zigzag ladder graph G, with n+1 nodes, and edges { i,i+1}, and {i,i+2}.  
[ Zaslavsky ],  [citeseer]

    

  •  Graphical zonotopes for zigzag ladder graph Z(G)
 [Postnikov]

    

  •  Voronoi cells of cographical lattice for zigzag ladder graphs (primary parallelohedra, primary parallelotopes) [F. Vallentin]
    

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 18, 54, 162, ...  2*3^n  acyclic orientations of zigzag ladder [ OEIS A008776 ]

    

  •  Number of Facets:
     0, 2, 6, 12, 20,  ... n^2+n directed edge cuts of the zigzag ladder on n+1 nodes [ OEIS A002378]

    		• 

    

    

    top     index 
     

    

Acyclotope (fan graph)

    


    

	  
    
 (0, 8, 0, 4)
    		
 

    

  •  Acyclotopes  A(F_1,n)  for fan graphs F_1,n.
[ Zaslavsky ],  [citeseer]

    

  •  Graphical zonotopes for fan graph Z(F_1,n)
 [Postnikov]

    

  •  Shuffle polytope of Path graph and Point graphical zonotopes. [ arxiv ] (Chapoton, Pilaud) 
    

    

  •  Voronoi cells of cographical lattice for fan graphs (primary parallelohedra, primary parallelotopes) [F. Vallentin]
    

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 18, 54, 162, ...  2*3^n  acyclic orientations of fan graph
     [ OEIS A025192 ]

    

  •  Number of Facets:
     0, 2, 6, 12, 20,  ... n^2+n directed edge cuts of the fan graph on n+1 nodes [ OEIS A002378]

    		• 

    

    

    top     index 
     

    

Multiplihedron

    


    

	 
    
 (0, 6, 2, 5)
    		
 

    

  •  Multiplihedra  J, M  
 [arxiv] (Forcey)

    

  •  Path-graph-multiplihedra  JP_n  
 [arxiv] (Devadoss, Forcey)
    

  •  2-associahedra W_n0  = W_0n for the sequences(n,0) or (0,n). [arxiv](N. Bottman) 

     

  •  Step 1 Biassociahedra  K^2_n  
[arxiv](M. Markl)
    

  •  (in low dimensions) trihomomorphism axioms (pasting diagrams) in tricategories [books] (Gordon, Power, Street)
    
Constrainahedra  C(2,n) = C(n,2),
[ arxiv ] ( Bottman, Poliakova)
        or  Constr(1,n) = Constr(n,1), [ arxiv ] (Chapoton, Pilaud)
    

  •  (1,n)-Multiplihedra Mult(1,n) [ arxiv ] Pilaud, Poliakova	
    	
	
    		• 




  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 21, 80, ...  Catalan transform of Catalan numbers  [ OEIS A121988]

    

  •  Number of Facets:
     0, 2, 6, 13, 25, 46,  ...  n(n + 1)/2+ 2^n - 1. [ OEIS ?][see: i]
    

    		• 

    

    

    top     index 
     

    

Path-graph Tubing polytope

    


    

	 
    
 (0, 6, 0, 7)
    		
 

    

  •  Path-graph Tubing polytope  T  

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 22, 90, ...  Large Schröder numbers  [ OEIS A006318]

    

  •  Number of Facets:
     0, 2, 6, 13,  ...   [ OEIS ?][see: i]
    

    		• 

    

    

    top     index 
     

    

(2,n)-Brick Polytope 

    


    

	 
    
 (0, 5, 2, 6)**
    		
 

    

  •  (2,n)-Brick Polytopes  B^2(n)
 [arxiv](V. Pilaud) 

    

  •  convex hulls of the brick vectors of all (2, n)-twists
    

  •  Brick Polytopes of the 2-kernels of 
    (size n) bubble sort networks Omega(B^2_(n+4))
 [arxiv](V. Pilaud, F. Santos)
    
• (2,n)-Beam Polytopes  Omega_(2,n)   [arxiv](V. Pilaud) 

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n-1
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 22, 92, 420, 2042, ...   [ OEIS A263791] and 
	[ OEIS conjecture A264868]

    

  •  Number of Facets:
     0, 2, 6, 13, 25, 45, 78, 132, ... [ OEIS  A065220]
    

  •  f-vectors: 
     1, 2, 1, 6, 6, 1, 22, 33, 13, 
    1, 92, 185, 118, 25, 1, 420, 1062, 945, 346, 45, ... [ OEIS ?]
    

    		• 

    

    

    top     index 
     

    


Pterahedron

    


    

	 
    
 (0, 5, 2, 6)
    		
 

    

  •  Pterahedra  P_t  
 [L. Berry] 

    

  •  fan-graph-associahedra [arxiv] (S. Devadoss)

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n-1
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 22, 94, 464, ...  Catalan transform of the factorials  [ OEIS ?]

    

  •  Number of Facets:
     0, 2, 6, 13, 25, 46,  ...  n(n + 1)/2+ 2^n - 1. [ OEIS ?][see: i]
    

  •  f-vectors: 
     1, 2, 1, 6, 6, 1, 22, 33, 13, 1, 94, ... [ OEIS ?]
    

    		• 

    

    

    top     index 
     

    




Permutohedron, permutahedron

    


    

	  
    
 (0, 6, 0, 8)
    		
 

    

  •  (Type A) Permutohedra/ permutahedra  P, S  
 [wiki] 

    

  •  Secondary polytopes of the prisms of simplices. [e-book] (Gelfand, Kapranov, Zelevinsky)
    

  •  complete-graph-associahedra [arxiv] (S. Devadoss)

    

  •  complete-graph-multiplihedra [arxiv] (Devadoss, Forcey)
	
  •  Path graph graphicahedra [arxiv] (Schulte et al.)	
		
  •  Star-graph operahedra (since the line graph of a star graph is complete) [arxiv] (Ward) [arxiv] (Laplante-Anfossi)

    

  •  Step 1 Bipermutohedra  P^n_m  
[arxiv](M. Markl)[arxiv](S.Saneblidze, R. Umble)  

    

  •  Step 1 Biassociahedra  K(n,m) = K^n_m  
 [arxiv](M. Markl)[arxiv](S.Saneblidze, R. Umble)
    

  •  Acyclotopes for complete graphs  A(K_n)  
[ Zaslavsky ],  [ citeseer ]

    

  •  Graphical zonotopes for complete graphs Z(K_n)
 [Postnikov]

    

  •  Zonotopes polar to the braid arrangements.
    

  •  Fiber polytopes of unit cubes over line segments. 
    

  •   Voronoi cells of cographical lattice for complete graphs (primary parallelohedra, primary parallelotopes) [F. Vallentin]
    

  •   Brillouin zone (Wigner-Seitz cell of reciprocal space) for Face Centered Cubic lattice in 3d [wiki]

    

  •   Poset-associahedra for antichain with minimal element adjoined (claw) [arxiv](Devadoss et.al.)
    

  •  (Galashin) poset associahedra for antichain with minimal element adjoined (claw) [arxiv] (P. Galashin)
	
    	

  •  Quotientopes  P , whose upper ideal of shards  contains all the shards.
[Pilaud, Santos]

    

  •   1-skeleton is Cayley graph for symmetric group, using transpositions.[wiki] 

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 6, 24, 120, ...  n!  [ OEIS A000142][see: i, ii, iii] 

    

  •  Number of Facets:
     0, 2, 6, 14, 30  ...  2^(n+1) -2  [ OEIS A000918]

    

  •  f-vectors: 
     1, 2, 1, 6, 6, 1, 24, 36, 14, 1, ... [ OEIS A019538]
    

  •  h-vectors: 
     1, 4, 1, 1, 11, 11, 1, 1, 26, 66, 26, 1, ... [ OEIS A008292]

    		• 

    

    

    top     index 
     

    

Type D permutohedron

    


    

	  
    
 (0, 6, 0, 8)
    		
 

    

  •  Type D permutohedra  PD  
 [zib.de] (Reiner, Ziegler) 

    

    		• 

  •  Dimensions: 
     2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     4, 24, 192, ...  2^(n-1)*n! [ OEIS A002866]

    

  •  Number of Facets:
     4, 14, 48,  ...  3^n - n*2^(n-1) - 1 [ OEIS ?] (Thanks N. Reading)

    

  •  f-vectors: 
      1, 4, 4, 1, 24, 36, 14, 1,  192,  384,  240,  48, 1,... [ OEIS A145902]

    		• 

    

    

    top     index 
     

    



Graph multiplihedron (cycle graph)

    


    

	  
    
(0, 6, 0, 8)**
    		
 

  •  Cycle-multiplihedra  JG  
 [arxiv] (Devadoss, Forcey)
    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 6, 24, 104, ... OPEN [ OEIS ?]

    

  •  Number of Facets:
     0, 2, 6, 14, 28,  ... OPEN [ OEIS ?]

    		• 

    

    

    top     index 
     

    

(3,n) Brick Polytope

    


    

	  
    
(0, 6, 0, 8)**
    		
 

    

  •  (3,n)-Brick Polytopes  B^3(n)
 [arxiv](V. Pilaud) 

    

  •  convex hulls of the brick vectors of all (3, n)-twists
    

  •  Brick Polytopes of the 3-kernels of 
    (size n) bubble sort networks Omega(B^3_n)
 [arxiv](V. Pilaud, F. Santos)
    

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 6, 24, 114, 612, 3600, ... [ OEIS A263791] and 
	[ OEIS ?]

    

  •  Number of Facets:
     0, 2, 6, 14, 29, 57, 109, 205, ... OPEN [ OEIS ?]

    		• 

    

    

    top     index 
     

    


 2-associahedron for sequence (0,n,0)

    


    

	  
    
(0, 6, 0, 8) conj. polytope
    		
 

    

  •  (0,n,0) 2-associahedra W_0n0 
 [arxiv](N. Bottman) 

    

  •  Faces are 2-tubings based on the sequence 0,n,0
    

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 6, 24, 108, 520, 2620, 13648, 72956, ... OPEN [ OEIS ?]

    

  •  Number of Facets:
     0, 2, 6, 14, 29, 57, 110, 212, ... OPEN [ OEIS ?]

    		• 

    

    

    top     index 
     

    

 Strong Rectangulotopes 

    


    

	  
    
(0, 6, 0, 8) 
    		
 

    

  • Strong Rectangulotopes  SR(n) 
 [ Cardinal, Pilaud] 
    

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 6, 24, 116, 642, 3938, 26194, ...OPEN [ OEIS ]

    

  •  Number of Facets:
     0, 2, 6, 14, 30, 62 ... 2^(n+1)-2 [ OEIS ?]

    		• 

    

    

    top     index 
     

    	


Diagonal rectangulation polytope

    


    

	 
    
(0, 6, 4, 4)
    		
 

    

  •  Diagonal rectangulation polytopes
 [arxiv] (Law, Reading)

    

  •  Quotientopes  P , whose upper ideal of shards  contains the basic shards, all upper shards,  and all lower shards.
[Pilaud, Santos]

    

  • Weak Rectangulotopes  WR(n)  [ Cardinal, Pilaud] 
    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 1, 2, 6, 22, 92, ... Baxter permutations  [ OEIS A001181]

    

  •  Number of Facets:
     0, 2, 6, 14, 30  ...  2^(n+1) -2  [ OEIS A000918]


    		• 

    

    

    top     index 
     

    

Ordered partial partition polytope

    


    

	 
    
(0, 6, 0, 9)
    		
 

    

  •  Ordered partial partition polytope (Determinant volume polytopes) F_A
 [arxiv] (R. Houston, A. Goucher, N. Johnston)

    

  •  Complete Graph Tubing polytopes, 
[A. Goucher]

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 1, 2, 6, 26, 150, 1082, ... , Sum_{k>=1} k^n/2^k. [ OEIS A000629]

    

  •  Number of Facets:
     0, 2, 6, 15, 36, 85, 198,  ...  n * (1 + 2^(n-1))  [ OEIS A215149]

    

  •  f-vectors: 
     1,
2, 1,
6, 6, 1,
26, 39, 15, 1,
150, 300, 186, 36, 1, ... 

    		• 

    

    

    top     index 
     

    

Cycle graph tubing polytope

    


    

	 
    
(0, 6, 0, 9)
    		
 

    

  •  Cycle Graph Tubing polytopes, 
[A. Goucher]

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 1, 2, 6, 26, 126, 642, ... [ OEIS ? ]

    

  •  Number of Facets:
     0, 2, 6, 15,  [ OEIS ? ]

    

  •  f-vectors: 
     1,
2, 1,
6, 6, 1,
26, 39, 15, 1,
126, ... 

    		• 

    

    

    top     index 
     

    

	
Type B permutohedron

    


    

	  
    
(0, 12, 0, 8, 0, 6)
    
	Image redrawn from Fig. 2.4 of [Fomin, Reading]     		
 

    

  •  Type B permutohedra  PB 
[arxiv] [Fomin, Reading]

    

  •  Conjectured: Acyclotopes of signed complete graphs [Zaslavsky]

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     1, 2, 8, 48, 384, ...  2^n*n! = (2n)!!  [ OEIS A000165][see: i] 

    

  •  Number of Facets:
     0, 2, 8, 26, 80, ... 3^n-1  [ OEIS A024023] (Thanks to N. Reading)

    

  •  f-vectors: 
     1, 2, 1, 8, 8, 1, 48, 72, 26, 1, ... [ OEIS A145901 (dual)]

    		• 


    

    

    top     index 
     

    

Type B Coxeter-associahedron

    


    

	  
    
(0, 36, 0, 0, 0, 6, 0, 0, 0, 8)
    
Image redrawn from Fig. 3 of [Reiner, Ziegler],
    
	original credited to Jürgen Richter-Gebert.
    		
 

    

  •  Type B Coxeter-associahedra  KPB 
[zib.de] [Reiner, Ziegler]

    

    		• 


  •  Dimensions: 
     1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope: 
     2, 8, 96,  ...  2^n*n!*(Catalan number)  [ OEIS conjecture]

    

  •  Number of Facets:
     2, 8, 50, ...  OPEN  [ OEIS ?]

    

    		• 


    

    

    top     index 
     

    

Type A Coxeter-associahedron (Permutoassociahedron)

    


    

	  
    
(0, 42, 24, 0, 0, 0, 0, 0, 0, 8)
    
	Image redrawn from Fig. 4 of [Reiner, Ziegler],
    
	original credited to Jürgen Richter-Gebert.
    		
 

    

  •  Type A Coxeter-associahedra  KPA 
[zib.de] [Reiner, Ziegler] 

    

  •  Permutoassociahedra, Permuto-associahedra KP  
 [M. Batanin, via R. Street] 

    

    		• 

  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 12, 120, 1680... n!*(Catalan number) [ OEIS ]

    

  •  Number of Facets:
     0, 2, 12, 74, ...  Ordered Bell numbers -1  [ OEIS A000670][ OEIS A052875]

    

  •  f-vectors: 
     1, 2, 1, 12, 12, 1, 120, 192, 74, 1, ... 
[ OEIS ?]

    		• 

    

    

    top     index 
     

    


Type D Coxeter-associahedra 

    


    

	  
    
(0, 18, 0, 4, 0, 0, 0, 0, 0, 4)
    
	Image redrawn from Fig. 3 of [Reiner, Ziegler],
    
	original credited to Jürgen Richter-Gebert.
    		
 

    

  •  Type D Coxeter-associahedra  KPD 
[zib.de] [Reiner, Ziegler]

    

    		• 


  •  Dimensions: 
     0, 1, 2, 3, ... n
    

  •  Number of Vertices in nth polytope:
     1, 2, 4, 48, ...  [ OEIS ? ]

    

  •  Number of Facets:
     0, 2, 4, 26, ... [ OEIS ?]

    

  •  f-vectors: 
     1, 2, 1, 4, 4, 1, 48, 72, 26, 1, ... 

    		• 

    

    

    top     index 
     

    

(Polars of) Even Dimensional Cyclic Polytopes with d+3 vertices 

    


    

	  
    
(No 3d term)
    
    		
 

    

  •  (Polars of) C(2k+3,2k) Alternating Cyclic Polytopes 
    

  •  Multiassociahedra: k-triangulations of the (2k+3)-gon
     
   [ arxiv ] ( V. Pilaud )
    

  •  (Polars of) Amplituhedra A(d+3,1,d) 
     
    the alternating polytopes of dimension d with d+3 
     
    vertices (in projective space)[ arxiv ]
     
    ( S. Karp, L. Williams)
    

    		• 


  •  Dimensions: 
     2, 4, 6, 8,... 2k 
    

  •  Number of Vertices in kth polytope: 
     5, 14, 30, 55, 91, 140, 204   ... ((k+3) choose 3)+((k+2) choose 3)   [OEIS A000330 ]

    

  •  Number of Facets:
     5, 7, 9, 11... 2k+3 
    

  •  f-vectors: 
     1, 5, 5, 1, 14, 28, 21, 7, 1, 30, 90, 117, 84, 36, 9, 1, 55, 220, 407, 451, 330, 165, 55, 11, 1...

    		• 

    

    

    top     index 
     

    



Symmetric Path Polytope

    


    

	  
    
(No 3d term)
    
[polymake for n=4,5]
    		
 

    

  •  s-t Path Polytopes of complete graph  Path(n)= Path(K_n), n>1  [Springer] (A. Schrijver)

    

  •  convex_hull({incidence_vector_F | F a path from node s to node t of the complete graph on n nodes})
    

    		• 


  •  Dimensions: 
     0, 1, 4, 8,... conject. (n choose 2)-2 [ OEIS A034856]
    

  •  Number of Vertices in nth polytope: 
     1, 2, 5, 16, 65,  ...  Sum_{k=0..(n-2)} (n-2)!/k!   [ OEIS A000522][see: i]

    

  •  Number of Facets:
     2, 5, 25, ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 5, 10, 10, 5, 1, 16, 102, 334, 622, 685, 442, 156, 25, 1... [ OEIS ?]

    		• 

    

    

    top     index 
     

    



Birkhoff polytope

    


    

	  
    
 
    
(No 3d term.)
    
[polymake for n=3]
    
[
	Sage library]
    		
 

    

  •  Birkhoff polytopes  B  [wiki]  

    

  •  assignment polytope
    

  •  Transportation polytope Trans_n(1, 1, . . . , 1) [arxiv] (Mészáros, Morales, Rhoades)
    

  •  perfect matching polytope of complete bipartite graph 
    

  •  set of doubly stochastic matrices 
    

  •  convex_hull({M | M an nxn  permutation matrix}) 
    

    		• 

  •  Dimensions: 
     0, 1, 4, 9, 16 ... (n-1)^2
    

  •  Number of Vertices in nth polytope:
    1, 2, 6, 24, ... n! [ OEIS A000142][see: i, ii, iii] 

    

  •  Number of Facets:
     0, 2, 9, 16,... n^2 [ OEIS A000290]
    

  •  f-vectors: 
     1, 2, 1, 6, 15, 18, 9, 1 ... [ OEIS ?]

    		• 

    

    

    top     index 
     

    


Alternating Sign Matrix Polytope 

    


    

	  
    
(No 3d term.)
    
[polymake for n=3]
    		
 

    

  •  Alternating Sign Matrix polytopes  ASM(n) [arXiv] (J. Striker) 

    

  •  convex_hull({char_vector_ASM | ASM an nxn  alternating sign matrix }) 
    

    		• 


  •  Dimensions: 
     0, 1, 4, 9, 16, ... (n-1)^2
    

  •  Number of Vertices in nth polytope:
    1, 2, 7, 42, 429, 7436,... Product[j=0..n-1](3j+1)!/(n+j)!  [ OEIS A005130]

    

  •  Number of Facets:
     0, 2, 4, 8, 20, 40, 68, 104,...,4[(n-2)^2 +1][ OEIS A128445] 
    

  •  f-vectors: 
     1, 2, 1, 7, 17, 18, 8, 1, 42,  ... [ OEIS ?]

    		• 

    

    

    top     index 
     

    



Linear signed order polytope

    


    

	  
    
(No 3d term)
    
[polymake for n=2]
    		
 

    

  •  Linear signed ordering polytopes  Q  [science direct] (S. Fiorini, P. Fishburn)

    

  •  convex_hull({char_vector_SLO | SLO a signed linear order with 2n elements}) 
    

    		• 


  •  Dimensions: 
     0, 1, 4, 9, 16, ... n^2
    

  •  Number of Vertices in nth polytope:
    1, 2, 8, 48, 384 ... 2^n*n!=(2n)!!  [ OEIS A000165][see: i] 

    

  •  Number of Facets:
     0, 2, 16, 82, 8480,  ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 8, 24, 32, 16, 1  ... [ OEIS ?]

    		• 

    

    

    top     index 
     

    

Huffman Polytope, Huffmanhedron

    


    

	  
    
[polymake for n=4]
    		
 

    

  •  Huffmanhedra, Huffman polytopes HP(n) [ResearchGate] (J. Maurras, T. Nguyen, V. Nguyen)DOI: 10.1016/j.dam.2012.05.004)

    

  •  convex_hull({char_vector_t | t a Huffman tree with n leaves}) 
    

    		• 


  •  Dimensions: 
     0, 2, 4, 5, 6, ... n
    

  •  Number of Vertices in nth polytope:
    1, 1, 3, 13, 75, ... OPEN [ OEIS ?]
    

  •  Number of Facets:
     0, 3, 9, ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 13, 30, 26, 9, 1, ... [ OEIS ?]

    		• 

    

    

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Perfect Matching polytope of complete graph

    


    

	  
    
(No 3d term.)
    
[polymake for 2n=6]
    		
 

    

  •  Perfect Matching polytope of complete graph on 2n nodes  PM(n) = PM(K_2n)    [wiki]

    

  •  convex_hull({incidence_vector_PM | PM a perfect matching of the complete graph on 2n nodes}) [PNAS](P.Diaconis, S. Holmes)
    

    

    		• 

  •  Dimensions: 
     0, 2, 9, ... 
    

  •  Number of Vertices in nth polytope:
    1, 3, 15, 105,  ... (2n-1)!! [ OEIS A001147][see: i]
    

  •  Number of Facets:
     0, 3, 25, ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 15, 105, 435, 1095, 1657, 1470, 735, 195, 25, 1, ... [ OEIS ?]

    		• 

    

    

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Interval order polytope

    


    

	  
    
(No 3d term.)
    		
 

    

  •  Interval order polytopes  P_IO  [citeseer] (R. Muller, A. Schulz) [wiki]

    

  •  Interval order polytopes  P_IO(D_n)  of the complete digraph.
    

  •  convex_hull({char_vector_IO | IO an interval order with n elements}) 
    

    		• 

  •  Dimensions: 
     0, 2, 6, ... 
    

  •  Number of Vertices in nth polytope:
    1, 3, 19, 207, 3451, ... [ OEIS ]
    

  •  Number of Facets:
     0, 3, 17, ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 19, 96, 193, 183, 84, 17, 1, 207, ... [ OEIS ?]

    		• 

    

    

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Partial order polytope

    


    

	  
    
(No 3d term.)
    		
 

    

  •  Partial order polytopes  P_PO  [science direct] (S. Fiorini)

    

  •  convex_hull({char_vector_PO | PO a partial order with n elements}) 
    

    		• 

  •  Dimensions: 
     0, 2, 6, 12, ... n(n-1)
    

  •  Number of Vertices in nth polytope:
    1, 3, 19, 219, ... OPEN [ OEIS ?]
    

  •  Number of Facets:
     0, 3, 17, 128  ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 19, 96, 193, 183, 84, 17, 1, 219, 5791 ... [ OEIS ?]

    		• 

    

    

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Symmetric Traveling salesman polytope

    


    

	  
    
(No 3d term.)
    		
 

    

  •  Symmetric Traveling Salesman polytopes  STSP  [zib.de] 
	(M. Grötschel, M. Padberg), [SMAPO library]
	

    

  •  convex_hull({char_vector_HC | HC a Hamiltonian cycle of the complete graph on n nodes}) 
    		• 


  •  Dimensions: 
     0, 2, 5, 9, 14 ... n(n-3)/2
    

  •  Number of Vertices in nth polytope:
    1, 3, 12, 60, ... (n-1)!/2[ OEIS A001710]
    

  •  Number of Facets:
     0, 3, 20 ,100, 3437, 194187, 42104442,... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 12, 60, 120, 90, 20, 1
 ... [ OEIS ?]

    		• 

    

    

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Asymmetric Traveling Salesman polytope 

    


    
 (No 2d,3d,4d terms.)


 

    

  •  Asymmetric Traveling salesman polytopes  ATSP(n) 
[cornell] (L. Billera, A. Sarangarajan)
     [science direct] (R. Euler,H. Le Verge)

    

  •  convex_hull({char_vector_HC | HC a Hamiltonian cycle of the complete digraph on n nodes}) 
     
    		• 


  •  Dimensions: 
     1, 5, 11, 19 ... n(n-3)+1;  n>2
    

  •  Number of Vertices in polytope for n nodes:
    2, 6, 24, 120, ... (n-1)![ OEIS A000142][see: i, ii, iii] 
    

  •  Number of Facets:
     2, 6, 390, 319015,... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 2, 1, 6, 15, 20, 15, 6, 1, 24, ... [ OEIS ?]

    		• 

    

    

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Splitohedron

    


    

	  
    
(No 3d term.)
    
[polymake for n=5,6]
    
[MatLab code for running PolySplit]
    
[also need matrix generator]
    
[and    branch and bound algorithm]
    
[and    distance algorithm]
    
[and    RF-metric algorithm]
    
    		
 

    

  •  Splitohedra  Sp_n  [arxiv] (S. Forcey, L. Keefe, W. Sands)

    

  •  relaxation of the  Balanced Minimum Evolution Polytope BME(n).
    

  •  intersection of half-spaces{split-facets, intersecting cherry facets, 
	
    caterpillar facets and the cherry clade-faces    } from BME(n)
     and also obeying the 
	{Kraft equalities}. 
    

    		• 

  •  Dimensions (start n =3): 
     0, 2, 5, 9, 14 ... (n choose 2)-n
    

  •  Number of Vertices in nth polytope:
    1, 3, 27, 2335, ... OPEN [ OEIS ?]
    

  •  Number of Facets:
     0, 3, 40, 85, 161 ... (2^n+n^3-3n^2-2)/2 [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 27, 165, 310, 210, 40, 1, 2335, ... [ OEIS ?]

    		• 

    

    

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Balanced Minimum Evolution Polytope

    


    

	  
    
(No 3d term.)
    
[polymake for n=5,6]
    
[faces and facets](2 page pdf.)
    		
 

    

  •  BME polytopes  P_n  = BME(n) 
	[arxiv] 
	(D. Haws, T. Hodge, R. Yoshida)
    [almob]
	(Eickmeyer et. al.)
	[blog resources]

  •  Balanced Minimal Evolution polytopes [arxiv], [arxiv](Forcey et. al.)

    

  •  convex_hull({dist_vector_T | T a binary tree with n labeled leaves}) 
    

    		• 

  •  Dimensions (start n =3): 
     0, 2, 5, 9, 14 ... (n choose 2)-n
    

  •  Number of Vertices in nth polytope:
    1, 3, 15, 105,  ... (2n-5)!! [ OEIS A001147][see: i]
    

  •  Number of Facets:
     0, 3, 52, 90262... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 15, 105, 250, 210, 52, 1, 105, 5460... [ OEIS ?]

    		• 

    

    

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Acyclic subgraph polytope

    


    

	  
    
(No 3d term.)
    
[polymake for n=3]
    		
 

    

  •  Acyclic subgraph polytopes P_AC  [ULB] (S. Fiorini)
    
[springer] (M. Grötschel, M. Jünger, G. Reinelt),
     [MIT] (M. Goemans, L. Hall)

    

  •  convex_hull({char_vector_AC | AC an acyclic subgraph of complete digraph on n nodes}) 
    

  •  Dicycle covering polytope P_DC [ULB] (S. Fiorini)
    

  •  convex_hull({<1,...,1> - char_vector_AC | AC an acyclic subgraph of complete digraph on n nodes}) 
    

    		• 


  •  Dimensions: 
     0, 2, 6, 12, ... n(n-1)
    

  •  Number of Vertices in nth polytope:
    1, 3, 25, 543, 29281,  ... [ OEIS A003024]
     

  •  Number of Facets:
     0, 3, 11 ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 25, 93, 142, 111, 48, 11, 1, ... [ OEIS ?]

    		• 

    

    

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Weak order polytope

    


    

	  
    
(No 3d term.)
    		
 

    

  •  Weak order polytopes  P_WO  [science direct] (S. Fiorini, P. Fishburn)

    

  •  convex_hull({char_vector_WO | WO a weak order with n elements}) 
    

    		• 

  •  Dimensions: 
     0, 2, 6, 12, ... n(n-1)
    

  •  Number of Vertices in nth polytope:
    1, 3, 13, 75, ... sum{k=0..inf} (k^n)/(2^(k-1))[ OEIS A000670]
    

  •  Number of Facets:
     0, 3, 15, 106  ... OPEN [ OEIS ?]
    

  •  f-vectors: 
     1, 3, 3, 1, 13, ... [ OEIS ?]

    		• 

    

    

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coming: Graphical Traveling salesman polytope  GTSP(n) [SMAPO library]

    




 
** 3d term is simple but probably not later terms.

    top     index 
     




    
ARRAYS (two or more indices) (we are trying to decide how to organize these!) 

    
Full entry coming soon:

    
  
	


    

	  
    
 (0, 22, 0, 0, 8)
    		

 


    
Cyclic polytopes C(n,m)  ...  
[ wiki] 

    
Pairahedra Ph(n,m)  ...  (0, 4, 4, 3)   I(2,1)= I(1,2)  [T. Tradler],[ T. Tradler] 


    
Finite product lattice polytopes P_(l,m,...,k)  ...  (0, 4, 4, 3)    P_(1,2)
    [J. Bloom] 


    
Resultohedra  (indexed by trees)  ... (0,4,4,4); (0, 7, 6, 6) [M. Batanin, via R. Street] [arxiv](Batanin)


    
(Step 2)     Biassociahedra  KK(n,m) = B^n_m    ...  KK(2,3) = KK(3,2)= heptagon   [arxiv](M. Markl)
     [arxiv](S.Saneblidze, R. Umble)  


  ...   KK(2,4) = KK(4,2)  =
(0, 13, 3, 0, 5), 32 vert. 21 facets 
    [arxiv](M. Markl)  
    
 KK(3,3)  =
(0, 22, 0, 0, 8), 44 vert. 30 facets 
     [arxiv](M. Markl) 
     


    
Bimultiplihedra  JJ(n,m) 
      ...   JJ(2,2) = octagon ;  JJ(2,3)  =(0, 19, 0, 3, 4, 0, 0, 2) 46 vert.  28 facets  [arxiv] (S.Saneblidze, R. Umble)


    
(Step 2) Bipermutahedra  PP(n,m) 
      ...  PP(1,2) = heptagon; PP(2,2) = KK(3,3)  =
(0, 22, 0, 0, 8), 44 vert. 30 facets; [arxiv](S.Saneblidze, R. Umble)  


    
Operahedra  G. Laplante-Anfossi 

    
Shuffles of deformed polytopes  F. Chapoton,  Pilaud 

    
Higher Secondary polytopes.  Galashin, Postnikov, Williams

    
Galashin's Poset  associahedra  P. Galashin 

    	
Bridge polytopes  L. Williams 	
	
	
    	
Constrainahedra  N. Bottman, D. Poliakova 	
	
    
Ordered partial partition polytopes  R. Houston, A. P. Goucher, N. Johnston 	
	
    


		
    		






    

    


    

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INDEX
    




    
	

    
 # |


 A |


 B |


 C |


 D |


 E |


 F |


 G |


 H |


 I |


 J |


 K |


 L |


 M |


 N |


 O |


 P |


 Q |


 R |


 S |


 T |


 U |


 V |


 W |


 X |


 Y |


 Z |



    

    



    



 


2-associahedra


 

Acyclic subgraph polytope
    

Acyclotopes: see Graphical Zonotopes
    

Acyclotope (cycle graph)
    

Alternating Sign Matrix Polytope 
    

Amplituhedron for biadjoint scalar phi^3 theory 
    

Asymmetric Traveling Salesman polytope  
    

Associahedron 


    

Amplituhedra


 

Balanced Minimum Evolution Polytope
    

Beam Polytope
    
	
Biassociahedra
      

Bimultiplihedra
     
 
Bipartite Subgraph polytope
    

Birkhoff polytope

    

Brick Polytopes



Bridge polytopes 

    

 

Cayley Polytopes
    	

Chain Polytopes





Composihedron 

    
	
Constrainahedra
	
	
	

Coxeter Associahedra and Permutahedra
Cube
    Cubeahedron(edgeless graph)
    Cut Polytope
    Cyclic polytopes
    Cyclohedron
    Demihypercube
    Diagonal rectangulation polytope
    Edgeless-graph-associahedra

    Finite product lattice polytopes 
    Flow polytopes 
    Freehedron

    



 Graph Associahedra 

 Graph Composihedra 

Graph Cubeahedra 


 Graph Multiplihedra Graphical Traveling Salesman polytope 

    

 Graphical Zonotopes Graph composihedron (cycle graph)
    Graph composihedron (edgeless graph)
    Graph multiplihedron (cycle graph)
    Graph multiplihedron (edgeless graph)
    Halohedron
    Higher Secondary polytopes
    Huffman Polytope
    Hypercube
    Independent set polytope of Uniform matroid
    Interval order polytope
    Linear Ordering Polytope
    Linear signed order polytope

    



Matching Polytopes



Multiassociahedra ((k-crossing)-free diagonalizations of n-gon)

Multiplihedron

    Operahedra
    Ordered partial partition polytopes
    
	
Order Polytopes

Pairahedra
    Partial order polytope
    Path-graph Tubing polytope
    Path polytope
    

Perfect Matching Polytopes:  see Matching Polytopes.
    Permutohedron/ permutahedron
    Permutoassociahedron

    

Poset Associahedra
Pterahedron

    
 

Quotientopes


    
Rectangulotopes	
Resultohedra
    Simplex

    


Secondary Polytopes





Shuffle Polytopes
	
Splitohedron

    Stanley-Pitman Polytope
    Stellohedron
    Symmetric Traveling Salesman polytope

    

Traveling Salesman Polytopes
	

Tubing Polytopes
	Tutte Polytope
    Type A Coxeter-associahedron (Permutoassociahedron)
    Type B Coxeter-associahedron
    Type B permutohedron
    Type D Associahedron
    Type D Coxeter-associahedra 
    Type D permutohedron
    Vertex cover polytope: complete graph

    Weak order polytope
    

Zonotopes: see Graphical Zonotopes
    top     index 
    
Conjectured

    
simplicial complex of  k-triangulations  or k-tubings .

    
Species compositions:  Painted tree polytopes .

    
pseudograph-multiplihedra, poset multiplihedra.


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