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A007578 Number of Young tableaux of height <= 7.
(Formerly M1219)
11
1, 1, 2, 4, 10, 26, 76, 232, 763, 2611, 9415, 35135, 136335, 544623, 2242618, 9463508, 40917803, 180620411, 813405580, 3728248990, 17377551032, 82232982872, 394742985974, 1919885633178, 9453682648281, 47086636037601, 237071351741426, 1205689994416252 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also the number of n-length words w over 7-ary alphabet {a1,a2,...,a7} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a7), 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) ~ 45/32 * 7^(n+21/2)/(Pi^(3/2)*n^(21/2)). - 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, 7, []):

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],

       ((4*n^3+78*n^2+424*n+495)*a(n-1)

       +(n-1)*(34*n^2+280*n+305)*a(n-2)

       -2*(n-1)*(n-2)*(38*n+145)*a(n-3)

       -105*(n-1)*(n-2)*(n-3)*a(n-4)) /

       ((n+6)*(n+10)*(n+12)))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

MATHEMATICA

RecurrenceTable[{105 (-3+n) (-2+n) (-1+n) a[-4+n]+2 (-2+n) (-1+n) (145+38 n) a[-3+n]-(-1+n) (305+280 n+34 n^2) a[-2+n]+(-495-424 n-78 n^2-4 n^3) a[-1+n]+(6+n) (10+n) (12+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=7 of A182172. - Alois P. Heinz, May 30 2012

Sequence in context: A239078 A007123 A220871 * A239079 A007580 A239080

Adjacent sequences:  A007575 A007576 A007577 * A007579 A007580 A007581

KEYWORD

nonn

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Alois P. Heinz, Apr 10 2012

STATUS

approved

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Last modified July 22 00:13 EDT 2017. Contains 289648 sequences.