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

Author

Geoffrey Critzer, Aug 18 2020

Keywords

• nonn
• tabf

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).

Examples

			  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
		

Crossrefs

Cf. A052141, A316674.

Programs

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