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A318289 Number of standard Young tableaux of 2n cells and height >= n.

%I A318289 #27 Sep 08 2021 12:39:07
%S A318289 1,2,9,56,441,3952,40161,442248,5288933,67100072,904408398,
%T A318289 12777826272,189324035423,2917525618256,46754429476800,
%U A318289 774965979970096,13279872426589125,234395323126241080,4258775222885983350,79442662095373693728,1520453631213137081776
%N A318289 Number of standard Young tableaux of 2n cells and height >= n.
%C A318289 Also number of self-inverse permutations of [2n] with longest increasing subsequence of length >= n.
%H A318289 Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>
%H A318289 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A318289 a(n) = A182222(2n,n).
%t A318289 h[l_] := Module[{n = Length[l]}, Total[l]!/Product[Product[1+l[[i]]-j+Sum[ If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
%t A318289 g[n_, i_, l_] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g[n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
%t A318289 t[n_, k_] := g[n, n, {}] - If[k == 0, 0, g[n, k-1, {}]];
%t A318289 a[n_] := a[n] = t[2n, n];
%t A318289 Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* _Jean-François Alcover_, Sep 08 2021, after _Alois P. Heinz_ in A182222 *)
%Y A318289 Cf. A182222.
%K A318289 nonn
%O A318289 0,2
%A A318289 _Alois P. Heinz_, Nov 04 2018