A392457 Number of vertices with distinct coordinates up to permutation in the Loday realization of the (n-1)-dimensional associahedron.

1, 1, 2, 3, 6, 11, 23, 45, 95, 195, 417, 865, 1877, 4001, 8687, 18653, 40650, 87989, 192360

Offset

1,3

Comments

The (n-1)-dimensional associahedron is a polytope whose skeleton is the Hasse diagram of the Tamari lattice on binary trees with n internal nodes (which has C_n elements).

For any binary tree T, corresponding coordinates in the Loday realization are obtained by multiplying for each internal node of T the number of leaves of its left child and the number of leaves of its right child.

Isomorphic binary trees have the same coordinates up to permutation. More precisely, binary trees giving rectangle tilings with the same shapes (cf. A247139) have the same coordinates up to permutation.

Links

Table of n, a(n) for n=1..19.

Jean-Louis Loday, Realization of the Stasheff polytope, arXiv:math/0212126 [math.AT], 2002.

Ludovic Schwob, Middle orders: all distributive lattices between weak and Bruhat, arXiv:2606.12129 [math.CO], 2026. See p. 14 (Table 1).

Example

The a(4) = 3 Loday coordinates up to permutation are {4, 3, 2, 1}, {4, 4, 1, 1} and {6, 2, 2, 1}, which can be respectively obtained from the following binary trees:
            o                     o                     o
           / \                   / \                   / \
          o   \                 o   \                 /   \
         / \   \               / \   \               /     \
        o   \   \             /   \   \             o       \
       / \   \   \           /     \   \           / \       \
      o   \   \   \         o       o   \         o   \       o
     / \   \   \   \       / \     / \   \       / \   \     / \
    o   o   o   o   o     o   o   o   o   o     o   o   o   o   o

Crossrefs

Cf. A000108, A247139, A382440.

Sequence in context: A093304 A036590 A382440 * A247139 A392756 A036591

Adjacent sequences: A392454 A392455 A392456 * A392458 A392459 A392460

Keyword

nonn,more,changed

Author

Ludovic Schwob, Jan 13 2026

Status

approved