cp's OEIS Frontend

A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.

%I A350950 #5 Mar 18 2022 00:20:38
%S A350950 0,0,1,-1,0,0,1,0,0,-1,0,1,1,0,-1,-1,0,1,1,0,0,1,0,0,-3,0,3,1,1,0,0,0,
%T A350950 -1,-1,-2,0,1,0,0,-1,0,1,1,0,2,-1,0,1,-2,-2,-1,1,1,2,-3,0,0,0,0,-1,1,
%U A350950 -1,3,-1,-2,0,0,0,-1,-1,1,1,0,0,0,1,-3,1,1,0
%N A350950 Number of even parts minus number of even conjugate parts in the integer partition with Heinz number n.
%C A350950 The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
%F A350950 a(n) = A257992(n) - A350847(n).
%F A350950 a(A122111(n)) = -a(n), where A122111 represents partition conjugation.
%e A350950 The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 2 - 1 = 1.
%t A350950 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350950 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350950 Table[Count[primeMS[n],_?EvenQ]-Count[conj[primeMS[n]],_?EvenQ],{n,100}]
%Y A350950 The version comparing even with odd parts is A195017.
%Y A350950 The version comparing even with odd conjugate parts is A350849.
%Y A350950 The version comparing even conjugate with odd conjugate parts is A350941.
%Y A350950 The version comparing odd with even conjugate parts is A350942.
%Y A350950 Positions of 0's are A350945, counted by A350948.
%Y A350950 The version comparing odd with odd conjugate parts is A350951.
%Y A350950 There are four individual statistics:
%Y A350950 - A257991 counts odd parts, conjugate A344616.
%Y A350950 - A257992 counts even parts, conjugate A350847.
%Y A350950 There are five other possible pairings of statistics:
%Y A350950 - A325698: # of even parts = # of odd parts, counted by A045931.
%Y A350950 - A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350950 - A350848: # of even conj parts = # of odd conj parts, counted by A045931.
%Y A350950 - A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%Y A350950 - A350944: # of odd parts = # of odd conjugate parts, counted by A277103.
%Y A350950 There are three possible double-pairings of statistics:
%Y A350950 - A350946, counted by A351977.
%Y A350950 - A350949, counted by A351976.
%Y A350950 - A351980, counted by A351981.
%Y A350950 The case of all four statistics equal is A350947, counted by A351978.
%Y A350950 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350950 A116482 counts partitions by number of even parts.
%Y A350950 A122111 represents partition conjugation using Heinz numbers.
%Y A350950 A316524 gives the alternating sum of prime indices.
%Y A350950 Cf. A028260, A088218, A098123, A236559, A239241, A241638, A325700, A347450.
%K A350950 sign
%O A350950 1,25
%A A350950 _Gus Wiseman_, Mar 14 2022