cp's OEIS Frontend

A351976 Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.

%I A351976 #8 Mar 18 2022 00:20:54
%S A351976 1,1,0,1,1,1,1,1,2,2,2,4,5,5,5,6,9,11,11,16,21,22,24,31,41,46,48,64,
%T A351976 82,91,98,120,155,175,188,237,297,329,357,437,544,607,658,803,987,
%U A351976 1098,1196,1432,1749,1955,2126,2541,3071,3417,3729,4406,5291,5890,6426
%N A351976 Number of integer partitions of n with (1) as many odd parts as odd conjugate parts and (2) as many even parts as even conjugate parts.
%e A351976 The a(n) partitions for selected n:
%e A351976 n = 3     8       11        12        15          16
%e A351976    ----------------------------------------------------------
%e A351976     (21)  (332)   (4322)    (4332)    (4443)      (4444)
%e A351976           (4211)  (4331)    (4422)    (54321)     (53332)
%e A351976                   (4421)    (4431)    (632211)    (55222)
%e A351976                   (611111)  (53211)   (633111)    (55411)
%e A351976                             (621111)  (642111)    (633211)
%e A351976                                       (81111111)  (642211)
%e A351976                                                   (643111)
%e A351976                                                   (7321111)
%e A351976                                                   (82111111)
%t A351976 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A351976 Table[Length[Select[IntegerPartitions[n],Count[#,_?OddQ]==Count[conj[#],_?OddQ]&&Count[#,_?EvenQ]==Count[conj[#],_?EvenQ]&]],{n,0,30}]
%Y A351976 The first condition alone is A277103, ranked by A350944, strict A000700.
%Y A351976 The second condition alone is A350948, ranked by A350945.
%Y A351976 These partitions are ranked by A350949.
%Y A351976 A000041 counts integer partitions.
%Y A351976 A122111 represents partition conjugation using Heinz numbers.
%Y A351976 A195017 = # of even parts - # of odd parts.
%Y A351976 There are four statistics:
%Y A351976 - A257991 = # of odd parts, conjugate A344616.
%Y A351976 - A257992 = # of even parts, conjugate A350847.
%Y A351976 There are four other possible pairings of statistics:
%Y A351976 - A045931: # even = # odd, ranked by A325698, strict A239241.
%Y A351976 - A045931: # even conj = # odd conj, ranked by A350848, strict A352129.
%Y A351976 - A277579: # even = # odd conj, ranked by A349157, strict A352131.
%Y A351976 - A277579: # even conj = # odd, ranked by A350943, strict A352130.
%Y A351976 There are two other possible double-pairings of statistics:
%Y A351976 - A351977: # even = # odd, # even conj = # odd conj, ranked by A350946.
%Y A351976 - A351981: # even = # odd conj, # odd = # even conj, ranked by A351980.
%Y A351976 The case of all four statistics equal is A351978, ranked by A350947.
%Y A351976 Cf. A088218, A098123, A130780, A171966, A236559, A236914, A241638, A350849, A350941, A350942, A350950, A350951.
%K A351976 nonn
%O A351976 0,9
%A A351976 _Gus Wiseman_, Mar 14 2022