cp's OEIS Frontend

A300121 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.

%I A300121 #8 May 24 2018 16:10:05
%S A300121 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 A300121 144,69,86,512,105,1024,16,208,320,209,82,2048,704,512,86,4096,318,
%U A300121 8192,216,262,1536,16384,64,465,262,1232,528,32768,209,588,245,2912,3328
%N A300121 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 A300121 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 A300121 The a(9) = 11 tableaux:
%e A300121 1 1
%e A300121 1 1
%e A300121 .
%e A300121 2 1   1 1   1 1   1 2
%e A300121 1 1   1 2   2 2   1 2
%e A300121 .
%e A300121 1 1   1 2   1 2   1 3
%e A300121 2 3   1 3   3 3   2 3
%e A300121 .
%e A300121 1 2   1 3
%e A300121 3 4   2 4
%t A300121 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 A300121 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 A300121 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A300121 Table[Length[cos[Reverse[primeMS[n]]]],{n,50}]
%Y A300121 Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A296561, A297388, A299699, A299925, A299926, A300056, A300060, A300118, A300120, A300122, A300123, A300124.
%K A300121 nonn
%O A300121 1,3
%A A300121 _Gus Wiseman_, Feb 25 2018