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COMMENTS
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Number of standard Young tableaux of n cells and <= 10 columns.
Also the number of n-length words w over 10-ary alphabet {a1,a2,...,a10} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a10), where #(z,x) counts the number of letters x in word z.
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MAPLE
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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, 10, []):
seq (a(n), n=0..30);
# second Maple program
a:= proc(n) option remember;
`if`(n<6, [1, 1, 2, 4, 10, 26][n+1],
((70*n^4+4144*n^3+84986*n^2+685800*n+1656000)*a(n-1)
+4*(n-1)*(35*n^4+1778*n^3+30106*n^2+184221*n+244350)*a(n-2)
-8*(n-1)*(n-2)*(518*n^2+11916*n+59265)*a(n-3)
-16*(n-1)*(n-2)*(n-3)*(259*n^2+4819*n+17355)*a(n-4)
+21600*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)
+14400*(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)) /
((n+21)*(n+9)*(n+16)*(n+25)*(n+24)))
end:
seq (a(n), n=0..30); # Alois P. Heinz, Oct 12 2012
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