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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.

%I A337192 #16 Sep 18 2020 17:28:52
%S A337192 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,
%T A337192 1875,2751,2570,1490,490,70,1,36,405,2290,7755,17088,25493,26070,
%U A337192 18060,8120,2142,252,1,49,735,5537,25235,76293,160867,242845,264936,207690,114282,41958,9240,924
%N 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.
%C A337192 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.
%C A337192 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).
%e A337192   1,
%e A337192   1, 1,
%e A337192   1, 4,  5,   2,
%e A337192   1, 9,  27,  37,  24,   6,
%e A337192   1, 16, 84,  216, 309,  252,  110,  20,
%e A337192   1, 25, 200, 800, 1875, 2751, 2570, 1490, 490, 70
%t A337192 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}]),
%t A337192    z], {1}] // Grid
%Y A337192 Cf. A052141, A316674.
%K A337192 nonn,tabf
%O A337192 0,5
%A A337192 _Geoffrey Critzer_, Aug 18 2020