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A218264 Number of standard Young tableaux of n cells and height >= 4.

%I A218264 #23 Aug 23 2021 05:49:40
%S A218264 1,5,25,105,441,1785,7308,29898,124641,526669,2276846,10038964,
%T A218264 45353269,209442533,990777442,4791502156,23707812077,119810145337,
%U A218264 618483875689,3256714122209,17488997849803,95671400358075,532959538382100,3020603738202750,17411069344112895
%N A218264 Number of standard Young tableaux of n cells and height >= 4.
%C A218264 Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 4. a(4)=1: 1234; a(5)=5: 12345, 12354, 12435, 13245, 21345.
%H A218264 Alois P. Heinz, <a href="/A218264/b218264.txt">Table of n, a(n) for n = 4..800</a>
%H A218264 Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>
%H A218264 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A218264 a(n) = A000085(n) - A001006(n) = A182172(n,n) - A182172(n,3).
%p A218264 a:= proc(n) option remember;
%p A218264       `if`(n<5, [0$4, 1][n+1], ((-5-7*n+3*n^2)*a(n-1)
%p A218264         +(n-1)*(n^2-n-11)*a(n-2) -2*n*(n-1)*(n-2)*a(n-3)
%p A218264         -3*(n-1)*(n-2)*(n-3)*a(n-4))/((n+2)*(n-4)))
%p A218264     end:
%p A218264 seq(a(n), n=4..30);
%t A218264 a[n_] := a[n] = If[n<5, {0,0,0,0,1}[[n+1]], ((-5-7n+3n^2)a[n-1] + (n-1)(n^2-n-11)a[n-2] - 2n(n-1)(n-2)a[n-3] - 3(n-1)(n-2)(n-3)a[n-4])/ ((n+2)(n-4))];
%t A218264 Table[a[n], {n, 4, 30}] (* _Jean-François Alcover_, Aug 23 2021, after _Alois P. Heinz_ *)
%Y A218264 Column k=4 of A182222.
%K A218264 nonn
%O A218264 4,2
%A A218264 _Alois P. Heinz_, Oct 24 2012