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 A007580 Number of Young tableaux of height <= 8. (Formerly M1220) 6

%I M1220

%S 1,2,4,10,26,76,232,764,2619,9486,35596,139392,562848,2352064,

%T 10092160,44546320,201158620,930213752,4387327088,21115314916,

%U 103386386516,515097746072,2605341147472,13378787264584,69622529312665,367161088308490,1959294979429380

%N Number of Young tableaux of height <= 8.

%C 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 number of letters x in word z. - _Alois P. Heinz_, May 30 2012

%D 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.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A007580/b007580.txt">Table of n, a(n) for n = 1..400</a>

%H <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>

%p h:= proc(l) local n; n:=nops(l);

%p k=i+1..n), j=1..l[i]), i=1..n)

%p end:

%p g:= proc(n, i, l) option remember;

%p `if`(n=0, h(l), `if`(i=1, h([l[], 1\$n]), `if`(i<1, 0,

%p g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))

%p end:

%p a:= n-> g(n, 8, []):

%p seq (a(n), n=1..30); # _Alois P. Heinz_, Apr 10 2012

%p # second Maple program

%p a:= proc(n) option remember;

%p `if`(n<4, [1, 1, 2, 4][n+1],

%p ((40*n^3+1084*n^2+8684*n+18480)*a(n-1)

%p +16*(n-1)*(5*n^3+107*n^2+610*n+600)*a(n-2)

%p -1024*(n-1)*(n-2)*(n+6)*a(n-3)

%p -1024*(n-1)*(n-2)*(n-3)*(n+4)*a(n-4)) /

%p ((n+7)*(n+12)*(n+15)*(n+16)))

%p end:

%p seq (a(n), n=1..30); # _Alois P. Heinz_, Oct 12 2012

%Y Column k=8 of A182172. - _Alois P. Heinz_, May 30 2012

%K nonn

%O 1,2

%A _Simon Plouffe_

%E More terms from _Alois P. Heinz_, Apr 10 2012

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