A007578 Number of Young tableaux of height <= 7.

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

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

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

• 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
• Andrei Asinowski, Axel Bacher, Cyril Banderier, Bernhard Gittenberger, Analytic combinatorics of lattice paths with forbidden patterns, the vectorial kernel method, and generating functions for pushdown automata, Laboratoire d'Informatique de Paris Nord (LIPN 2019).
• 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.
• 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.
• Index entries for sequences related to Young tableaux.

Crossrefs

Column k=7 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, 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 *)

Formula

a(n) ~ 45/32 * 7^(n+21/2)/(Pi^(3/2)*n^(21/2)). - Vaclav Kotesovec, Sep 11 2013

Extensions

More terms from Alois P. Heinz, Apr 10 2012