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Volume of the (n-1)-dimensional associahedron in the Loday realization, multiplied by (n-1)!.
1

%I #15 Mar 23 2021 15:48:04

%S 1,1,7,142,5895,417201,45046558,6891812712,1417730229765,

%T 377158121463025

%N Volume of the (n-1)-dimensional associahedron in the Loday realization, multiplied by (n-1)!.

%C Here the volume is relative to the unit cell of the lattice which is the intersection of Z^n with the hyperplane spanning the polytope.

%H Alexander Postnikov, <a href="https://doi.org/10.1093/imrn/rnn153">Permutohedra, Associahedra, and Beyond</a>, International Mathematics Research Notices, 2009, 1026-1106; arXiv:<a href="https://arxiv.org/abs/math/0507163">math/0507163</a> [math.CO], 2005. See Example 10.3.

%t a[n_] := With[{npr = Subsets[Span @@ Range@n, {2}]}, Sum[With[{ip = Ordering@p}, Total[-p[[Table[Min@ip[[ij]], {ij, npr}]]]]^(n - 1)] / Times @@ Differences[p], {p, Permutations@Range@n}]];

%t Table[a[n], {n, 8}]

%Y Cf. A087422, A227414, A342811.

%K nonn,more

%O 1,3

%A _Andrey Zabolotskiy_, Mar 22 2021