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A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.

%I A352131 #5 Mar 18 2022 00:21:58
%S A352131 1,0,0,1,0,1,1,1,0,1,2,1,1,2,3,2,2,3,4,3,4,5,5,5,6,7,7,8,10,10,10,12,
%T A352131 14,15,14,17,21,20,20,25,28,28,29,34,39,39,40,47,52,53,56,64,70,71,77,
%U A352131 86,92,97,104,114,122
%N A352131 Number of strict integer partitions of n with same number of even parts as odd conjugate parts.
%e A352131 The a(n) strict partitions for selected n:
%e A352131 n = 3      10         14         18         21             24
%e A352131    ----------------------------------------------------------------------
%e A352131     (2,1)  (6,4)      (8,6)      (10,8)     (11,10)        (8,7,5,4)
%e A352131            (4,3,2,1)  (5,4,3,2)  (6,5,4,3)  (8,6,4,3)      (9,8,4,3)
%e A352131                       (6,5,2,1)  (7,6,3,2)  (8,7,4,2)      (10,8,4,2)
%e A352131                                  (8,7,2,1)  (10,8,2,1)     (10,9,3,2)
%e A352131                                             (6,5,4,3,2,1)  (11,10,2,1)
%e A352131                                                            (8,6,4,3,2,1)
%t A352131 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A352131 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Count[#,_?EvenQ]==Count[conj[#],_?OddQ]&]],{n,0,30}]
%Y A352131 This is the strict case of A277579, ranked by A349157 (zeros of A350849).
%Y A352131 The conjugate version is A352130, non-strict A277579 (ranked by A350943).
%Y A352131 A000041 counts integer partitions, strict A000009.
%Y A352131 A130780 counts partitions with no more even than odd parts, strict A239243.
%Y A352131 A171966 counts partitions with no more odd than even parts, strict A239240.
%Y A352131 There are four statistics:
%Y A352131 - A257991 = # of odd parts, conjugate A344616.
%Y A352131 - A257992 = # of even parts, conjugate A350847.
%Y A352131 There are four other pairings of statistics:
%Y A352131 - A045931, ranked by A325698, strict A239241.
%Y A352131 - A045931, ranked by A350848, strict A352129.
%Y A352131 - A277103, ranked by A350944.
%Y A352131 - A350948, ranked by A350945.
%Y A352131 There are three double-pairings of statistics:
%Y A352131 - A351976, ranked by A350949.
%Y A352131 - A351977, ranked by A350946, strict A352128.
%Y A352131 - A351981, ranked by A351980.
%Y A352131 The case of all four statistics equal is A351978, ranked by A350947.
%Y A352131 Cf. A027187, A027193, A103919, A122111, A236559, A325039, A344607, A344651, A345196, A350942, A350950, A350951.
%K A352131 nonn
%O A352131 0,11
%A A352131 _Gus Wiseman_, Mar 15 2022