A300122 Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.

1, 4, 13, 51, 183, 771, 3087, 13601, 59933, 278797, 1311719, 6453606, 32179898, 166075956, 871713213, 4704669005, 25831172649, 145260890323

Offset

1,2

Comments

The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns. 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

Table of n, a(n) for n=1..18.

Example

The a(3) = 13 tableaux:
1 1 1   1 1 2   1 2 2   1 2 3
.
1 1   1 1   1 2   1 2   1 3
1     2     1     3     2
.
1   1   1   1
1   1   2   2
1   2   2   3

Mathematica

undcon[y_]:=Select[Tuples[Range[0, #]&/@y], Function[v, GreaterEqual@@v&&With[{r=Select[Range[Length[y]], y[[#]]=!=v[[#]]&]}, Or[Length[r]<=1, And@@Table[v[[i]]<y[[i+1]], {i, Range[Min@@r, Max@@r-1]}]]]]];
cos[y_]:=cos[y]=With[{samples=Most[undcon[y]]}, If[Length[samples]===0, If[Total[y]===0, {{}}, {}], Join@@Table[Prepend[#, y]&/@cos[samples[[k]]], {k, 1, Length[samples]}]]];
Table[Sum[Length[cos[y]], {y, IntegerPartitions[n]}], {n, 12}]

Crossrefs

Cf. A000085, A000898, A006958, A138178, A238690, A259479, A259480, A296561, A297388, A299699, A299925, A299926, A300118, A300120, A300121, A300123, A300124.

Sequence in context: A149459 A149460 A219579 * A149461 A056276 A144035

Adjacent sequences: A300119 A300120 A300121 * A300123 A300124 A300125

Keyword

nonn,more

Author

Gus Wiseman, Feb 25 2018

Status

approved