A299968 Number of normal generalized Young tableaux of size n with all rows and columns strictly increasing.

1, 1, 2, 5, 15, 51, 189, 753, 3248, 14738, 70658, 354178, 1857703, 10121033, 57224955, 334321008, 2017234773, 12530668585, 80083779383, 525284893144, 3533663143981, 24336720018666, 171484380988738, 1234596183001927, 9075879776056533, 68052896425955296

Offset

0,3

Comments

A generalized Young tableau of shape y is an array obtained by replacing the dots in the Ferrers diagram of y with positive integers. A tableau is normal if its entries span an initial interval of positive integers.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..50 (first 46 terms from Ludovic Schwob)

D. E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific Journal of Mathematics, Vol. 34, No. 3 (1970), 709-727.

Wikipedia, Young tableau

Formula

a(n) = Sum_{k=0..n} 2^k * A238121(n,k). - Ludovic Schwob, Sep 23 2023

Example

The a(4) = 15 tableaux:
1 2 3 4
.
1 2 3   1 2 4   1 3 4   1 2 3   1 2 3
4       3       2       2       3
.
1 2   1 3   1 2
3 4   2 4   2 3
.
1 2   1 3   1 2   1 4   1 3
3     2     2     2     2
4     4     3     3     3
.
1
2
3
4

Mathematica

unddis[y_]:=DeleteCases[y-#, 0]&/@Tuples[Table[If[y[[i]]>Append[y, 0][[i+1]], {0, 1}, {0}], {i, Length[y]}]];
dos[y_]:=With[{sam=Rest[unddis[y]]}, If[Length[sam]===0, If[Total[y]===0, {{}}, {}], Join@@Table[Prepend[#, y]&/@dos[sam[[k]]], {k, 1, Length[sam]}]]];
Table[Sum[Length[dos[y]], {y, IntegerPartitions[n]}], {n, 1, 8}]

Crossrefs

Cf. A000085, A000898, A001221, A005117, A006958, A015128, A138178, A238121, A238690, A285175, A296561, A297388, A299926, A300120, A300122.

Sequence in context: A279554 A279555 A153197 * A279556 A108307 A275605

Adjacent sequences: A299965 A299966 A299967 * A299969 A299970 A299971

Keyword

nonn

Author

Gus Wiseman, Feb 26 2018

Extensions

More terms from Ludovic Schwob, Sep 23 2023

Status

approved