A215087 Number of solid standard Young tableaux of n cells and height <= 2.

1, 1, 3, 8, 26, 92, 372, 1566, 7086, 33550, 167504, 873226, 4764614, 26947632, 157926628, 954523378, 5945067490, 38060781922, 250345198424, 1688978186742, 11679437620552, 82652840640478, 598018846154666, 4418072084681592, 33298670603875846, 255782905412464810

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..40

S. B. Ekhad, D. Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux, arXiv:1202.6229v1 [math.CO], 2012

Wikipedia, Young tableau

Maple

b:= proc(n, l) option remember; `if`(n=0, 1,
       b(n-1, [l[], [1]])+ add(`if`(i=1 or nops(l[i])<nops(l[i-1]),
       b(n-1, subsop(i=[l[i][], 1], l)), 0)+ add(`if`(l[i][j]<2 and
       (i=1 or l[i][j]<l[i-1][j]) and (j=1 or l[i][j]<l[i][j-1]),
       b(n-1, subsop(i=subsop(j=l[i][j]+1, l[i]), l)), 0),
       j=1..nops(l[i])), i=1..nops(l)))
    end:
a:= n-> b(n, []):
seq(a(n), n=0..20);

Mathematica

b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, Append[l, {1}]] + Sum[If[i == 1 || Length[l[[i]]] < Length[l[[i - 1]]], b[n - 1, ReplacePart[l, i -> Append[l[[i]], 1]]], 0] + Sum[If[l[[i, j]] < 2 && (i == 1 || l[[i, j]] < l[[i - 1, j]]) && (j == 1 || l[[i, j]] < l[[i, j - 1]]), b[n - 1, ReplacePart[l, i -> ReplacePart[l[[i]], j -> l[[i, j]] + 1]]], 0], {j, 1, Length[l[[i]]]}], {i, 1, Length[l]}]];
a[n_] := b[n, {}];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 01 2023, after Alois P. Heinz *)

Crossrefs

Column k=2 of A215086.

Sequence in context: A148820 A148821 A074506 * A148822 A124528 A320344

Adjacent sequences: A215084 A215085 A215086 * A215088 A215089 A215090

Keyword

nonn

Author

Alois P. Heinz, Aug 02 2012

Status

approved