A300118 - OEIS

%I #8 May 24 2018 16:07:14

%S 1,2,3,3,4,4,5,4,6,5,6,5,7,6,7,5,8,7,9,6,8,7,10,6,10,8,10,7,11,8,12,6,

%T 9,9,11,8,13,10,10,7,14,9,15,8,11,11,16,7,15,11,11,9,17,11,12,8,12,12,

%U 18,9,19,13,12,7,13,10,20,10,13,12,21,9,22,14,15,11

%N Number of skew partitions whose quotient diagram is connected and whose numerator is the integer partition with Heinz number n.

%C The diagram of a connected skew partition is required to be connected as a polyomino but can have empty rows or columns.

%e The a(15) = 7 denominators are (), (1), (11), (22), (3), (31), (32) with diagrams:

%e o o o   . o o   . o o   . . o   . . .   . . .   o o o

%e o o     o o     . o     . .     o o     . o     o o

%e Missing are the two disconnected skew partitions:

%e . . o   . . o

%e o o     . o

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%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 Table[Length[undcon[Reverse[primeMS[n]]]],{n,100}]

%Y Cf. A000085, A000898, A056239, A006958, A138178, A153452, A238690, A259479, A259480, A296150, A297388, A299925, A299926, A300056, A300060, A300120, A300122, A300123, A300124.

%K nonn

%O 1,2

%A _Gus Wiseman_, Feb 25 2018