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A293128
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Number of standard Young tableaux of 2n cells and height <= n.
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4
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1, 1, 6, 51, 588, 7990, 126060, 2242618, 44546320, 977152266, 23500234512, 615372604033, 17442275104496, 532242021137346, 17399782340548920, 606732491690590816, 22477989291826848000, 881635273413199806994, 36493478646922003374096, 1589642562747880936613248
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OFFSET
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0,3
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COMMENTS
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Also the number of standard Young tableaux of 2n cells and <= n columns.
Also the number of 2n-length words w over n-ary alphabet {a1,a2,...,an} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,an), where #(z,x) counts the letters x in word z. The a(2) = 6 words of length 4 over alphabet {a,b} are: aaaa, aaab, aaba, abaa, aabb, abab.
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LINKS
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FORMULA
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MAPLE
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h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
<j, 0, 1), k=i+1..n), j=1..l[i]), i=1..n))(nops(l)):
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), add(
g(n-i*j, i-1, [l[], i$j]), j=0..n/i)):
a:= n-> g(2*n, n, []):
seq(a(n), n=0..15);
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MATHEMATICA
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h[l_] := With[{n = Length[l]}, Total[l]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] < j, 0, 1], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];
g[n_, i_, l_] := If[n == 0 || i == 1, h[Join[l, Table[1, {n}]]], Sum[g[n - i*j, i - 1, Join[l, Table[i, {j}]]], {j, 0, n/i}]];
a[n_] := g[2n, n, {}];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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