cp's OEIS Frontend

A350951 Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.

%I A350951 #5 Mar 18 2022 00:20:46
%S A350951 0,0,-2,2,-2,0,-4,2,0,0,-4,0,-6,-2,0,4,-6,0,-8,0,-2,-2,-8,2,2,-4,-2,
%T A350951 -2,-10,0,-10,4,-2,-4,0,2,-12,-6,-4,2,-12,-2,-14,-2,-2,-6,-14,2,0,2,
%U A350951 -4,-4,-16,0,0,0,-6,-8,-16,2,-18,-8,-4,6,-2,-2,-18,-4,-6,0
%N A350951 Number of odd parts minus number of odd conjugate parts in the integer partition with Heinz number n.
%C A350951 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.
%C A350951 All terms are even.
%F A350951 a(n) = A257991 - A344616(n).
%F A350951 a(A122111(n)) = -a(n), where A122111 represents partition conjugation.
%e A350951 The prime indices of 78 are (6,2,1), with conjugate (3,2,1,1,1,1), so a(78) = 1 - 5 = -4.
%t A350951 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A350951 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A350951 Table[Count[primeMS[n],_?OddQ]-Count[conj[primeMS[n]],_?OddQ],{n,100}]
%Y A350951 The version comparing even with odd parts is A195017.
%Y A350951 The version comparing even with odd conjugate parts is A350849.
%Y A350951 The version comparing even conjugate with odd conjugate parts is A350941.
%Y A350951 The version comparing odd with even conjugate parts is A350942.
%Y A350951 Positions of 0's are A350944, even rank case A345196, counted by A277103.
%Y A350951 The version comparing even with even conjugate parts is A350950.
%Y A350951 There are four individual statistics:
%Y A350951 - A257991 counts odd parts, conjugate A344616.
%Y A350951 - A257992 counts even parts, conjugate A350847.
%Y A350951 There are five other possible pairings of statistics:
%Y A350951 - A325698: # of even parts = # of odd parts, counted by A045931.
%Y A350951 - A349157: # of even parts = # of odd conjugate parts, counted by A277579.
%Y A350951 - A350848: # of even conj parts = # of odd conj parts, counted by A045931.
%Y A350951 - A350943: # of even conjugate parts = # of odd parts, counted by A277579.
%Y A350951 - A350945: # of even parts = # of even conjugate parts, counted by A350948.
%Y A350951 There are three possible double-pairings of statistics:
%Y A350951 - A350946, counted by A351977.
%Y A350951 - A350949, counted by A351976.
%Y A350951 - A351980, counted by A351981.
%Y A350951 The case of all four statistics equal is A350947, counted by A351978.
%Y A350951 A056239 adds up prime indices, counted by A001222, row sums of A112798.
%Y A350951 A103919 counts partitions by number of odd parts.
%Y A350951 A116482 counts partitions by number of even parts.
%Y A350951 A122111 represents partition conjugation using Heinz numbers.
%Y A350951 A316524 gives the alternating sum of prime indices.
%Y A350951 Cf. A026424, A028260, A053738, A098123, A130780, A171966, A236559, A236914, A239241, A241638, A325700.
%K A350951 sign
%O A350951 1,3
%A A350951 _Gus Wiseman_, Mar 14 2022