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A218263 Number of standard Young tableaux of n cells and height >= 3.

%I A218263 #26 Aug 23 2021 05:50:16
%S A218263 1,4,16,56,197,694,2494,9244,35234,139228,566788,2387048,10343101,
%T A218263 46193866,211775002,997265204,4809609062,23758479340,119952340180,
%U A218263 618883933480,3257842530546,17492187873444,95680438560276,532985197799976,3020676725917252
%N A218263 Number of standard Young tableaux of n cells and height >= 3.
%C A218263 Also number of self-inverse permutations in S_n with longest increasing subsequence of length >= 3. a(3)=1: 123; a(4)=4: 1234, 1243, 1324, 2134.
%H A218263 Alois P. Heinz, <a href="/A218263/b218263.txt">Table of n, a(n) for n = 3..800</a>
%H A218263 Wikipedia, <a href="https://en.wikipedia.org/wiki/Involution_(mathematics)">Involution (mathematics)</a>
%H A218263 Wikipedia, <a href="https://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>
%F A218263 a(n) = A000085(n) - A001405(n) = A182172(n,n) - A182172(n,2).
%F A218263 Conjecture: (n-6)*(n-3)*(n+1)*a(n) +(-n^3+6*n^2+11*n-36)*a(n-1) -(n-1)*(n^3-4*n^2-21*n+76)*a(n-2) +2*(n-1)*(n-2)*(3*n-19)*a(n-3) +4*(n-5)*(n-1)*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 04 2017
%p A218263 b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
%p A218263 a:= n-> b(n) -binomial(n, iquo(n, 2)):
%p A218263 seq(a(n), n=3..30);
%t A218263 b[n_] := b[n] = If[n<2, 1, b[n-1] + (n-1)*b[n-2]];
%t A218263 a[n_] := b[n] - Binomial[n, Quotient[n, 2]];
%t A218263 Table[a[n], {n, 3, 30}] (* _Jean-François Alcover_, Aug 23 2021, after _Alois P. Heinz_ *)
%Y A218263 Column k=3 of A182222.
%K A218263 nonn
%O A218263 3,2
%A A218263 _Alois P. Heinz_, Oct 24 2012