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

1, 4, 16, 56, 197, 694, 2494, 9244, 35234, 139228, 566788, 2387048, 10343101, 46193866, 211775002, 997265204, 4809609062, 23758479340, 119952340180, 618883933480, 3257842530546, 17492187873444, 95680438560276, 532985197799976, 3020676725917252

Offset: 3

Author

Alois P. Heinz, Oct 24 2012

Keywords

• nonn

Comments

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.

Links

• Alois P. Heinz, Table of n, a(n) for n = 3..800
• Wikipedia, Involution (mathematics)
• Wikipedia, Young tableau

Crossrefs

Column k=3 of A182222.

Programs

Maple

b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
a:= n-> b(n) -binomial(n, iquo(n, 2)):
seq(a(n), n=3..30);

Mathematica

b[n_] := b[n] = If[n<2, 1, b[n-1] + (n-1)*b[n-2]];
a[n_] := b[n] - Binomial[n, Quotient[n, 2]];
Table[a[n], {n, 3, 30}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

Formula

a(n) = A000085(n) - A001405(n) = A182172(n,n) - A182172(n,2).

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