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A007580 Number of Young tableaux of height <= 8.
(Formerly M1220)
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 (list; graph; refs; listen; history; text; internal format)
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

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.

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

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

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Last modified May 22 21:40 EDT 2017. Contains 286906 sequences.