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

1, 5, 25, 105, 441, 1785, 7308, 29898, 124641, 526669, 2276846, 10038964, 45353269, 209442533, 990777442, 4791502156, 23707812077, 119810145337, 618483875689, 3256714122209, 17488997849803, 95671400358075, 532959538382100, 3020603738202750, 17411069344112895

Offset: 4

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 >= 4. a(4)=1: 1234; a(5)=5: 12345, 12354, 12435, 13245, 21345.

Links

• Alois P. Heinz, Table of n, a(n) for n = 4..800
• Wikipedia, Involution (mathematics)
• Wikipedia, Young tableau

Crossrefs

Column k=4 of A182222.

Programs

Maple

a:= proc(n) option remember;
      `if`(n<5, [0$4, 1][n+1], ((-5-7*n+3*n^2)*a(n-1)
        +(n-1)*(n^2-n-11)*a(n-2) -2*n*(n-1)*(n-2)*a(n-3)
        -3*(n-1)*(n-2)*(n-3)*a(n-4))/((n+2)*(n-4)))
    end:
seq(a(n), n=4..30);

Mathematica

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))];
Table[a[n], {n, 4, 30}] (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

Formula

a(n) = A000085(n) - A001006(n) = A182172(n,n) - A182172(n,3).