A007579 Number of Young tableaux of height <= 6.

1, 1, 2, 4, 10, 26, 76, 231, 756, 2556, 9096, 33231, 126060, 488488, 1948232, 7907185, 32831370, 138321690, 593610420, 2579109780, 11377862340, 50726936820, 229078351992, 1043999256966, 4810194384348, 22340617618860, 104742353862360, 494547143860035

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

Comments

Also the number of n-length words w over 6-ary alphabet {a1,a2,...,a6} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a6), 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.
• Alon Regev, Amitai Regev, Doron Zeilberger, Identities in character tables of S_n, arXiv preprint arXiv:1507.03499 [math.CO], 2015.
• Index entries for sequences related to Young tableaux.

Crossrefs

Column k=6 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, 6, []):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 18 2012
# second Maple program:
a:= proc(n) option remember;
      `if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*a(n-1)
       +4*(n-1)*(10*n^2+58*n+33)*a(n-2) -144*(n-1)*(n-2)*a(n-3)
       -144*(n-1)*(n-2)*(n-3)*a(n-4))/ ((n+5)*(n+8)*(n+9)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Mathematica

RecurrenceTable[{144 (-3+n) (-2+n) (-1+n) a[-4+n]+144 (-2+n) (-1+n) a[-3+n]-4 (-1+n) (33+58 n+10 n^2) a[-2+n]-4 (84+46 n+5 n^2) a[-1+n]+(5+n) (8+n) (9+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) ~ 3/4 * 6^(n+15/2)/(Pi^(3/2)*n^(15/2)). - Vaclav Kotesovec, Sep 11 2013

D-finite with recurrence +(n+5)*(n+9)*(n+8)*a(n) +4*(-5*n^2-46*n-84)*a(n-1) -4*(n-1)*(10*n^2+58*n+33)*a(n-2) +144*(n-1)*(n-2)*a(n-3) +144*(n-1)*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 23 2021

Extensions

More terms from Alois P. Heinz, Apr 10 2012