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proposed

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proposed

h[l_] := With[{n = Length[l]}, Total[l]!/Product[ Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i + 1, n}], {j, 1, l[[i]]}], {i, 1, n}]];

b[n_, l_] := If[n < 1, h[l], b[n - 1, l] + b[n - 1, Append[l, n]]];

a[n_] := If[n == 0, 1, b[n - 1, {n}]];

Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 02 2022, after Alois P. Heinz *)

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Wikipedia, <a href="httphttps://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

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add (`(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)

seq ((a(n), n=0..12);

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editing

editing

approved

Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>

Wikipedia, <a href="http://en.wikipedia.org/wiki/Young_tableau">Young tableau</a>