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%I #24 Dec 16 2017 05:05:42
%S 1,1,2,4,10,26,76,232,764,2620,9495,35685,140031,567503,2382394,
%T 10290308,45780063,208852719,977152266,4674398032,22854255698,
%U 113957313538,579157509082,2995214721530,15752586526189,84145056172981,456221504976506,2508227921637772
%N Number of standard Young tableaux of n cells and height <= 9.
%C Number of standard Young tableaux of n cells and <= 9 columns.
%C Also the number of n-length words w over 9-ary alphabet {a1,a2,...,a9} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a9), where #(z,x) counts the letters x in word z.
%H Alois P. Heinz, <a href="/A212915/b212915.txt">Table of n, a(n) for n = 0..1000</a>
%H Juan B. Gil, Peter R. W. McNamara, Jordan O. Tirrell, Michael D. Weiner, <a href="https://arxiv.org/abs/1708.00513">From Dyck paths to standard Young tableaux</a>, arXiv:1708.00513 [math.CO], 2017.
%F a(n) ~ 14175/256 * 9^(n+18)/(Pi^2*n^18). - _Vaclav Kotesovec_, Sep 11 2013
%p h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j+
%p add(`if`(l[k]>=j, 1, 0), 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, 9, []):
%p seq(a(n), n=0..30);
%p # second Maple program:
%p a:= proc(n) option remember;
%p `if`(n<5, [1, 1, 2, 4, 10][n+1],
%p ((5*n^4+230*n^3+3574*n^2+20663*n+29393)*a(n-1)
%p +7*(n-1)*(10*n^3+266*n^2+1919*n+2713)*a(n-2)
%p -(n-1)*(n-2)*(230*n^2+3934*n+13587)*a(n-3)
%p -3*(n-1)*(n-2)*(n-3)*(263*n+1414)*a(n-4)
%p +945*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)) /
%p ((n+20)*(n+8)*(n+18)*(n+14)))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Oct 12 2012
%t Flatten[{1,RecurrenceTable[{-945 (-4+n) (-3+n) (-2+n) (-1+n) a[-5+n]+3 (-3+n) (-2+n) (-1+n) (1414+263 n) a[-4+n]+(-2+n) (-1+n) (13587+3934 n+230 n^2) a[-3+n]-7 (-1+n) (2713+1919 n+266 n^2+10 n^3) a[-2+n]+(-29393-20663 n-3574 n^2-230 n^3-5 n^4) a[-1+n]+(8+n) (14+n) (18+n) (20+n) a[n]==0,a[1]==1,a[2]==2,a[3]==4,a[4]==10,a[5]==26}, a, {n, 20}]}] (* _Vaclav Kotesovec_, Sep 11 2013 *)
%Y Column k=9 of A182172.
%K nonn
%O 0,3
%A _Alois P. Heinz_, May 30 2012
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