A212915 Number of standard Young tableaux of n cells and height <= 9.

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9495, 35685, 140031, 567503, 2382394, 10290308, 45780063, 208852719, 977152266, 4674398032, 22854255698, 113957313538, 579157509082, 2995214721530, 15752586526189, 84145056172981, 456221504976506, 2508227921637772

Offset: 0

Author

Alois P. Heinz, May 30 2012

Keywords

• nonn

Comments

Number of standard Young tableaux of n cells and <= 9 columns.

Also the number of n-length words w over 9-ary alphabet {a1,a2,...,a9} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a9), where #(z,x) counts the letters x in word z.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, From Dyck paths to standard Young tableaux, arXiv:1708.00513 [math.CO], 2017.

Crossrefs

Column k=9 of A182172.

Programs

Maple

h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
    end:
g:= proc(n, i, l) option remember;
      `if`(n=0, h(l), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 9, []):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<5, [1, 1, 2, 4, 10][n+1],
      ((5*n^4+230*n^3+3574*n^2+20663*n+29393)*a(n-1)
       +7*(n-1)*(10*n^3+266*n^2+1919*n+2713)*a(n-2)
       -(n-1)*(n-2)*(230*n^2+3934*n+13587)*a(n-3)
       -3*(n-1)*(n-2)*(n-3)*(263*n+1414)*a(n-4)
       +945*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)) /
       ((n+20)*(n+8)*(n+18)*(n+14)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Mathematica

Flatten[{1,RecurrenceTable[{-945 (-4+n) (-3+n) (-2+n) (-1+n) a[-5+n]+3 (-3+n) (-2+n) (-1+n) (1414+263 n) a[-4+n]+(-2+n) (-1+n) (13587+3934 n+230 n^2) a[-3+n]-7 (-1+n) (2713+1919 n+266 n^2+10 n^3) a[-2+n]+(-29393-20663 n-3574 n^2-230 n^3-5 n^4) a[-1+n]+(8+n) (14+n) (18+n) (20+n) a[n]==0,a[1]==1,a[2]==2,a[3]==4,a[4]==10,a[5]==26}, a, {n, 20}]}] (* Vaclav Kotesovec, Sep 11 2013 *)

Formula

a(n) ~ 14175/256 * 9^(n+18)/(Pi^2*n^18). - Vaclav Kotesovec, Sep 11 2013