A293128 Number of standard Young tableaux of 2n cells and height <= n.

1, 1, 6, 51, 588, 7990, 126060, 2242618, 44546320, 977152266, 23500234512, 615372604033, 17442275104496, 532242021137346, 17399782340548920, 606732491690590816, 22477989291826848000, 881635273413199806994, 36493478646922003374096, 1589642562747880936613248

Offset: 0

Author

Alois P. Heinz, Sep 30 2017

Keywords

• nonn

Comments

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.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..41 (terms 0..32 from Alois P. Heinz)
• Wikipedia, Young tableau

Crossrefs

Cf. A182172, A267436.

Programs

Maple

h:= l-> (n-> add(i, i=l)!/mul(mul(1+l[i]-j+add(`if`(l[k]
      `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);

Mathematica

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, {}];
a /@ Range[0, 15] (* Jean-François Alcover, Jan 02 2021, after Alois P. Heinz *)

Formula

a(n) = A182172(2n,n).