A007580 Number of Young tableaux of height <= 8. (Formerly M1220)

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

0,3

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.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A220871 A007578 A239079 * A239080 A212915 A239081

Adjacent sequences: A007577 A007578 A007579 * A007581 A007582 A007583

Keyword

nonn

Author

Simon Plouffe

Extensions

More terms from Alois P. Heinz, Apr 10 2012

Status

approved