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%I #8 May 24 2018 16:10:05
%S 1,1,2,2,4,5,8,4,11,12,16,12,32,28,31,8,64,31,128,33,82,64,256,28,69,
%T 144,69,86,512,105,1024,16,208,320,209,82,2048,704,512,86,4096,318,
%U 8192,216,262,1536,16384,64,465,262,1232,528,32768,209,588,245,2912,3328
%N 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 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. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%e The a(9) = 11 tableaux:
%e 1 1
%e 1 1
%e .
%e 2 1 1 1 1 1 1 2
%e 1 1 1 2 2 2 1 2
%e .
%e 1 1 1 2 1 2 1 3
%e 2 3 1 3 3 3 2 3
%e .
%e 1 2 1 3
%e 3 4 2 4
%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[{sam=Most[undcon[y]]},If[Length[sam]===0,If[Total[y]===0,{{}},{}],Join@@Table[Prepend[#,y]&/@cos[sam[[k]]],{k,1,Length[sam]}]]];
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]
%Y Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, A300122, A300123, A300124.
%K nonn
%O 1,3
%A _Gus Wiseman_, Feb 25 2018
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