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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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 17 15:12 EST 2020. Contains 330958 sequences. (Running on oeis4.)