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A300122
Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.
7
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.
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 25 2018
STATUS
approved