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A007579 Number of Young tableaux of height <= 6.

%I A007579 M1217 #61 Sep 23 2021 05:47:30
%S A007579 1,1,2,4,10,26,76,231,756,2556,9096,33231,126060,488488,1948232,
%T A007579 7907185,32831370,138321690,593610420,2579109780,11377862340,
%U A007579 50726936820,229078351992,1043999256966,4810194384348,22340617618860,104742353862360,494547143860035
%N A007579 Number of Young tableaux of height <= 6.
%C A007579 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
%D A007579 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007579 Alois P. Heinz, <a href="/A007579/b007579.txt">Table of n, a(n) for n = 0..1000</a>
%H A007579 F. Bergeron, L. Favreau and D. Krob, <a href="/A007578/a007578.pdf">Conjectures on the enumeration of tableaux of bounded height</a>, Preprint. (Annotated scanned copy)
%H A007579 F. Bergeron, L. Favreau and D. Krob, <a href="http://dx.doi.org/10.1016/0012-365X(94)00148-C">Conjectures on the enumeration of tableaux of bounded height</a>, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
%H A007579 Alon Regev, Amitai Regev, Doron Zeilberger, <a href="http://arxiv.org/abs/1507.03499">Identities in character tables of S_n</a>, arXiv preprint arXiv:1507.03499 [math.CO], 2015.
%H A007579 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F A007579 a(n) ~ 3/4 * 6^(n+15/2)/(Pi^(3/2)*n^(15/2)). - _Vaclav Kotesovec_, Sep 11 2013
%F A007579 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
%p A007579 h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
%p A007579       +add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
%p A007579     end:
%p A007579 g:= proc(n, i, l) option remember;
%p A007579       `if`(n=0, h(l), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
%p A007579         g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
%p A007579     end:
%p A007579 a:= n-> g(n, 6, []):
%p A007579 seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 18 2012
%p A007579 # second Maple program:
%p A007579 a:= proc(n) option remember;
%p A007579       `if`(n<4, [1, 1, 2, 4][n+1], ((20*n^2+184*n+336)*a(n-1)
%p A007579        +4*(n-1)*(10*n^2+58*n+33)*a(n-2) -144*(n-1)*(n-2)*a(n-3)
%p A007579        -144*(n-1)*(n-2)*(n-3)*a(n-4))/ ((n+5)*(n+8)*(n+9)))
%p A007579     end:
%p A007579 seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 12 2012
%t A007579 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 *)
%Y A007579 Column k=6 of A182172. - _Alois P. Heinz_, May 30 2012
%K A007579 nonn
%O A007579 0,3
%A A007579 _Simon Plouffe_
%E A007579 More terms from _Alois P. Heinz_, Apr 10 2012