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

Author

Gus Wiseman, Feb 26 2018

Keywords

• nonn

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.

Examples

			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
		

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

Crossrefs

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

Programs

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}]

Formula

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

Extensions

More terms from Ludovic Schwob, Sep 23 2023