A299966 Number of normal generalized Young tableaux, of shape the integer partition with Heinz number n, with all rows and columns weakly increasing and all regions non-singleton skew-partitions.

1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 3, 3, 5, 5, 5, 2, 8, 5, 13, 6, 13, 10, 21, 5, 11, 18, 11, 14, 34, 15, 55, 3, 26, 33, 23, 13, 89, 59, 54, 14, 144, 38, 233, 28, 31, 105, 377, 10, 47, 31, 106, 57, 610, 23, 60, 32, 206, 185, 987, 38, 1597, 324, 91, 5, 132, 93, 2584, 111

Offset: 1

Author

Gus Wiseman, Feb 22 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. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Examples

			The a(25) = 11 tableaux:
1 2 3   1 2 2   1 1 3   1 1 2
1 2 3   1 3 3   2 2 3   2 3 3
.
1 2 2   1 1 2   1 1 2   1 1 2   1 1 1   1 1 1
1 2 2   2 2 2   1 2 2   1 1 2   2 2 2   1 2 2
.
1 1 1
1 1 1
		

References

• Bruce E. Sagan, The Symmetric Group, Springer-Verlag New York, 2001.

Links

• Gus Wiseman, The a(30) = 15 non-singleton tableaux of shape (321).

Crossrefs

Cf. A000085, A056239, A063834, A112798, A122111, A138178, A153452, A238690, A296150, A296188, A297388, A299925, A299926, A299967.

Programs

Mathematica

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
undptns[y_]:=DeleteCases[Select[Tuples[Range[0,#]&/@y],OrderedQ[#,GreaterEqual]&],0,{2}];
eh[y_]:=If[Total[y]=!=1,1,0]+Sum[eh[c],{c,Select[undptns[y],Total[#]>1&&Total[y]-Total[#]>1&]}];
Table[eh[Reverse[primeMS[n]]],{n,60}]