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A316649 Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.

%I A316649 #28 Jul 27 2020 15:25:36
%S A316649 1,0,1,2,0,1,7,12,6,0,1,14,55,92,70,20,0,1,23,153,471,780,720,350,70,
%T A316649 0,1,34,336,1584,4251,7002,7238,4592,1638,252,0,1,47,640,4210,16175,
%U A316649 39733,65226,72660,54390,26250,7392,924,0,1,62,1107,9596,49225,164898,380731,623576,732618,614700,360162,140184,32604,3432
%N A316649 Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] ordered by the product order, 0 <= k <= 2n, n>=0.
%H A316649 Alois P. Heinz, <a href="/A316649/b316649.txt">Rows n = 0..100, flattened</a>
%e A316649 Triangle begins:
%e A316649   1;
%e A316649   0, 1,  2;
%e A316649   0, 1,  7,  12,    6;
%e A316649   0, 1, 14,  55,   92,   70,   20;
%e A316649   0, 1, 23, 153,  471,  780,  720,  350,   70;
%e A316649   0, 1, 34, 336, 1584, 4251, 7002, 7238, 4592, 1638, 252;
%e A316649   ...
%p A316649 b:= proc(n, m) option remember; expand(`if`(n+m=0, 1, add(add(
%p A316649      `if`(i+j=0, 0, b(sort([n-i, m-j])[])*x), j=0..m), i=0..n)))
%p A316649     end:
%p A316649 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A316649 seq(T(n), n=0..8);  # _Alois P. Heinz_, Jul 10 2018
%t A316649 Join[{{1}},Table[a =Sort[Level[Table[Table[{i, j}, {i, 0, n}], {j, 0, n}], {2}]];f[list1_, list2_] :=Boole[(list1 - list2)[[1]] < 1 \[And] (list1 - list2)[[2]] < 1];m = Table[Table[f[a[[l]], a[[k]]], {k, 1, Length[a]}], {l, 1, Length[a]}];Prepend[Table[
%t A316649      MatrixPower[m - IdentityMatrix[(n + 1)^2], k][[1, (n + 1)^2]], {k, 1, 2 n}], 0], {n, 1, 7}]] // Grid
%Y A316649 Columns k=0-2 give: A000007, A057427, A008865(n+1) for n>0.
%Y A316649 Row sums give A052141.
%Y A316649 T(n,n) gives A108628(n-1) for n>0.
%Y A316649 T(n,2n) gives A000984.
%Y A316649 Cf. A007318.
%K A316649 nonn,tabf
%O A316649 0,4
%A A316649 _Geoffrey Critzer_, Jul 09 2018