A337192 Triangular array read by rows. T(n,k) is the number of elements of rank k in the order complex of the poset P = [n] X [n], n=0, k=0 or n>0, 0<=k<=2n-1.

1, 1, 1, 1, 4, 5, 2, 1, 9, 27, 37, 24, 6, 1, 16, 84, 216, 309, 252, 110, 20, 1, 25, 200, 800, 1875, 2751, 2570, 1490, 490, 70, 1, 36, 405, 2290, 7755, 17088, 25493, 26070, 18060, 8120, 2142, 252, 1, 49, 735, 5537, 25235, 76293, 160867, 242845, 264936, 207690, 114282, 41958, 9240, 924

Offset

0,5

Comments

The poset P = [n] X [n] is the direct product of two chains of length n-1. The order complex of P is the set of all chains in P ordered by inclusion.

It appears that for n > 1, Sum_{k=0..2n-1} T(n,k) = 4*A052141(n-1). More generally, it appears that the number of elements in the order complex of [n]^k is four times the number of chains from bottom to top in [n]^k (Cf. A316674).

Links

Table of n, a(n) for n=0..56.

Example

  1,
  1, 1,
  1, 4,  5,   2,
  1, 9,  27,  37,  24,   6,
  1, 16, 84,  216, 309,  252,  110,  20,
  1, 25, 200, 800, 1875, 2751, 2570, 1490, 490, 70

Mathematica

f[x_, y_] := If[x <= y, 1, 0]; Prepend[CoefficientList[ 1 + z (Table[G = Array[f, {n, n}]; \[Zeta] = Level[Table[Table[Flatten[TensorProduct[G, G][[i]][[All, j]]], {j, 1, n}], {i, 1, n}], {2}]; a = Inverse[IdentityMatrix[n^2] - z (\[Zeta] - IdentityMatrix[n^2])]; Table[1, {n^2}].a.Table[1, {n^2}], {n, 1, 10}]),
   z], {1}] // Grid

Crossrefs

Cf. A052141, A316674.

Sequence in context: A336199 A267095 A016715 * A255701 A085548 A329957

Adjacent sequences: A337189 A337190 A337191 * A337193 A337194 A337195

Keyword

nonn,tabf

Author

Geoffrey Critzer, Aug 18 2020

Status

approved