A007580 Number of Young tableaux of height <= 8.

1, 1, 2, 4, 10, 26, 76, 232, 764, 2619, 9486, 35596, 139392, 562848, 2352064, 10092160, 44546320, 201158620, 930213752, 4387327088, 21115314916, 103386386516, 515097746072, 2605341147472, 13378787264584, 69622529312665, 367161088308490, 1959294979429380

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

Comments

Also the number of n-length words w over 8-ary alphabet {a1,a2,...,a8} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a8), where #(z,x) counts the letters x in word z. - Alois P. Heinz, May 30 2012

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Preprint. (Annotated scanned copy)
• F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
• Index entries for sequences related to Young tableaux.

Crossrefs

Column k=8 of A182172. - Alois P. Heinz, May 30 2012

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, 8, []):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 10 2012
# second Maple program:
a:= proc(n) option remember;
      `if`(n<4, [1, 1, 2, 4][n+1],
       ((40*n^3+1084*n^2+8684*n+18480)*a(n-1)
       +16*(n-1)*(5*n^3+107*n^2+610*n+600)*a(n-2)
       -1024*(n-1)*(n-2)*(n+6)*a(n-3)
       -1024*(n-1)*(n-2)*(n-3)*(n+4)*a(n-4)) /
       ((n+7)*(n+12)*(n+15)*(n+16)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Mathematica

RecurrenceTable[{1024 (-3+n) (-2+n) (-1+n) (4+n) a[-4+n]+1024 (-2+n) (-1+n) (6+n) a[-3+n]-16 (-1+n) (600+610 n+107 n^2+5 n^3) a[-2+n]-4 (4620+2171 n+271 n^2+10 n^3) a[-1+n]+(7+n) (12+n) (15+n) (16+n) a[n]==0,a[1]==1,a[2]==2,a[3]==4,a[4]==10}, a, {n, 20}] (* Vaclav Kotesovec, Sep 11 2013 *)

Formula

a(n) ~ 135/16 * 8^(n+14)/(Pi^2*n^14). - Vaclav Kotesovec, Sep 11 2013

Extensions

More terms from Alois P. Heinz, Apr 10 2012