A392457 - OEIS

A392457

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

%I #19 Jun 15 2026 10:12:29

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

%T 192360

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

%C 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).

%C 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.

%C 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.

%H Jean-Louis Loday, <a href="https://arxiv.org/abs/math/0212126">Realization of the Stasheff polytope</a>, arXiv:math/0212126 [math.AT], 2002.

%H Ludovic Schwob, <a href="https://arxiv.org/abs/2606.12129">Middle orders: all distributive lattices between weak and Bruhat</a>, arXiv:2606.12129 [math.CO], 2026. See p. 14 (Table 1).

%e 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:

%e o o o

%e / \ / \ / \

%e o \ o \ / \

%e / \ \ / \ \ / \

%e o \ \ / \ \ o \

%e / \ \ \ / \ \ / \ \

%e o \ \ \ o o \ o \ o

%e / \ \ \ \ / \ / \ \ / \ \ / \

%e o o o o o o o o o o o o o o o

%Y Cf. A000108, A247139, A382440.

%K nonn,more,changed

%O 1,3

%A _Ludovic Schwob_, Jan 13 2026