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%I A351981 #6 Mar 18 2022 00:21:24
%S A351981 1,0,0,1,0,0,0,0,1,2,1,0,2,2,2,4,2,1,6,8,7,9,13,14,15,19,21,23,32,40,
%T A351981 41,45,66,81,80,96,124,139,160,194,221,246,303,360,390,446,546,634,
%U A351981 703,810,971,1115,1250,1448,1685,1910
%N A351981 Number of integer partitions of n with as many even parts as odd conjugate parts, and as many odd parts as even conjugate parts.
%e A351981 The a(n) partitions for selected n:
%e A351981 n = 3    9      15       18       19       20         21
%e A351981    -----------------------------------------------------------
%e A351981     21   4221   622221   633222   633322   644321     643332
%e A351981          4311   632211   643221   643321   653321     654321
%e A351981                 642111   643311   644221   654221     665211
%e A351981                 651111   644211   644311   654311     82222221
%e A351981                          653211   653221   82222211   83222211
%e A351981                          663111   653311   84221111   84222111
%e A351981                                   654211   86111111   85221111
%e A351981                                   664111              86211111
%e A351981                                                       87111111
%e A351981 For example, the partition (6,6,3,1,1,1) has conjugate (6,3,3,2,2,2), and has 2 even, 4 odd, 4 even conjugate, and 2 odd conjugate parts, so is counted under a(18).
%t A351981 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A351981 Table[Length[Select[IntegerPartitions[n],Count[#,_?EvenQ]==Count[conj[#],_?OddQ]&&Count[#,_?OddQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y A351981 The first condition alone is A277579, ranked by A349157.
%Y A351981 The second condition alone is A277579, ranked by A350943.
%Y A351981 These partitions are ranked by A351980.
%Y A351981 There are four statistics:
%Y A351981 - A257991 = # of odd parts, conjugate A344616.
%Y A351981 - A257992 = # of even parts, conjugate A350847.
%Y A351981 There are four other pairings of statistics:
%Y A351981 - A045931: # of even parts = # of odd parts:
%Y A351981   - conjugate also A045931
%Y A351981   - ordered A098123
%Y A351981   - strict A239241
%Y A351981   - ranked by A325698
%Y A351981   - conjugate ranked by A350848
%Y A351981 - A277103: # of odd parts = # of odd conjugate parts, ranked by A350944.
%Y A351981 - A350948: # of even parts = # of even conjugate parts, ranked by A350945.
%Y A351981 There are two other double-pairings of statistics:
%Y A351981 - A351976, ranked by A350949.
%Y A351981 - A351977, ranked by A350946.
%Y A351981 The case of all four statistics equal is A351978, ranked by A350947.
%Y A351981 Cf. A000070, A088218, A122111, A130780, A171966, A195017, A236559, A236914, A350849, A350942.
%K A351981 nonn
%O A351981 0,10
%A A351981 _Gus Wiseman_, Mar 15 2022

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