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Number of normal generalized Young tableaux of size n with all rows and columns weakly increasing and all regions connected skew partitions.
7

%I #7 May 24 2018 16:04:54

%S 1,4,13,51,183,771,3087,13601,59933,278797,1311719,6453606,32179898,

%T 166075956,871713213,4704669005,25831172649,145260890323

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

%C 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.

%e The a(3) = 13 tableaux:

%e 1 1 1 1 1 2 1 2 2 1 2 3

%e .

%e 1 1 1 1 1 2 1 2 1 3

%e 1 2 1 3 2

%e .

%e 1 1 1 1

%e 1 1 2 2

%e 1 2 2 3

%t 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]}]]]]];

%t 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]}]]];

%t Table[Sum[Length[cos[y]],{y,IntegerPartitions[n]}],{n,12}]

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

%K nonn,more

%O 1,2

%A _Gus Wiseman_, Feb 25 2018