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Encyclopedia of Combinatorial Polytope Sequences...
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Searching the table:
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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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index

    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 ?]
  • top    index


    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 | 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

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    Conjectured
    simplicial complex of k-triangulations
    Species compositions: permutohedra with associahedra
    pseudograph-multiplihedra
     Back to research page. 

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