A349512 a(n) = binomial(n^3 + 3*n^2 - 3*n + 1, n^3).

1, 2, 6435, 4154246671960, 5397234129638871133346507775, 80240648651400365471854502514501453704175376562496, 54198670627270688013781273396239242514947489935351300645194042280183395324517200

Offset: 0

Author

Stefano Spezia, Nov 20 2021

Keywords

• nonn

Comments

a(n) is a sharp upper bound of the number of vertices of the polytope of the n X n X n stochastic tensors, or equivalently, of the number of Latin squares of order n, or equivalently, of the number of n X n X n line-stochastic (0,1)-tensors (see Zhang et al.).

Links

• Fuzhen Zhang and Xiao-Dong Zhang, Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach, Optimization, 69:4, 729-741, (2020). arXiv:2008.04655 [math.CO], 2020. See p. 6.
• Fuzhen Zhang and Xiao-Dong Zhang, Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors, arXiv:2110.12337 [math.CO], 2021. See p. 5.

Crossrefs

Cf. A000578, A229013.

Cf. A349506, A349507, A349508, A349509, A349510, A349511.

Programs

Mathematica

a[n_]:=Binomial[n^3+3n^2-3n+1,n^3]; Array[a,8,0]

Formula

A349508(n)/A349509(n) <= A349510(n) < A349511(n) < a(n) (see Corollary 7 in Zhang et al., 2021).

a(n) ~ C*3^(3(n - n^2))*exp(3*(3*n/2 + n^2))*n^(3(-n + n^2)), where C = e^(-15)/sqrt(54*Pi).