A218268 Number of standard Young tableaux of n cells and height >= 8.

1, 9, 81, 561, 3817, 23881, 147862, 886028, 5288933, 31178901, 183908244, 1081452450, 6381113064, 37719710024, 224141652938, 1337958249446, 8038507929319, 48593807722975, 295913856459150, 1814986751559300, 11220842616565050, 69921225307663290

Offset: 8

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 >= 8. a(8)=1: 12345678; a(9)=9: 123456789, 123456798, 123456879, 123457689, 123465789, 123546789, 124356789, 132456789, 213456789.

Links

• Alois P. Heinz, Table of n, a(n) for n = 8..300
• Wikipedia, Involution (mathematics)
• Wikipedia, Young tableau

Crossrefs

Column k=8 of A182222.

Programs

Maple

b:= proc(n) b(n):= `if`(n<2, 1, b(n-1) +(n-1)*b(n-2)) end:
g:= proc(n) option remember; `if`(n<4, [1, 1, 2, 4][n+1],
      ((4*n^3+78*n^2+424*n+495)*g(n-1) +(n-1)*(34*n^2+280*n
      +305)*g(n-2) -2*(n-1)*(n-2)*(38*n+145)*g(n-3) -105*(n-1)
      *(n-2)*(n-3)*g(n-4)) / ((n+6)*(n+10)*(n+12)))
    end:
a:= n-> b(n) -g(n):
seq(a(n), n=8..30);

Formula

a(n) = A000085(n) - A007578(n) = A182172(n,n) - A182172(n,7).