cp's OEIS Frontend

A349149 Number of even-length integer partitions of n with at most one odd part in the conjugate partition.

%I A349149 #8 Nov 12 2021 22:31:54
%S A349149 1,0,1,1,2,2,3,4,5,7,7,12,11,19,15,30,22,45,30,67,42,97,56,139,77,195,
%T A349149 101,272,135,373,176,508,231,684,297,915,385,1212,490,1597,627,2087,
%U A349149 792,2714,1002,3506,1255,4508,1575,5763,1958,7338,2436,9296,3010,11732
%N A349149 Number of even-length integer partitions of n with at most one odd part in the conjugate partition.
%C A349149 The alternating sum of a partition is equal to the number of odd parts in the conjugate partition, so this sequence counts even-length partitions with alternating sum <= 1.
%F A349149 a(2n) = A000041(n).
%F A349149 a(2n+1) = A000070(n-1).
%e A349149 The a(2) = 1 through a(9) = 7 partitions:
%e A349149   11   21   22     32     33       43       44         54
%e A349149             1111   2111   2211     2221     2222       3222
%e A349149                           111111   3211     3311       3321
%e A349149                                    211111   221111     4311
%e A349149                                             11111111   222111
%e A349149                                                        321111
%e A349149                                                        21111111
%t A349149 conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
%t A349149 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[conj[#],_?OddQ]<=1&]],{n,0,30}]
%Y A349149 The case of 0 odd conjugate parts is A000041 up to 0's, ranked by A000290.
%Y A349149 The case of 1 odd conjugate part is A000070 up to 0's.
%Y A349149 Even bisection of A100824, ranked by A349150.
%Y A349149 Ranked by A349151 /\ A028260.
%Y A349149 A045931 counts partitions with as many even as odd parts, ranked by A325698.
%Y A349149 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A349149 A122111 is a representation of partition conjugation.
%Y A349149 A277103 counts partitions with the same alternating sum as their conjugate.
%Y A349149 A277579 counts partitions with as many even parts as odd conjugate parts.
%Y A349149 A325039 counts partitions with the same product as their conjugate.
%Y A349149 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A349149 A345196 counts partitions with the same rev-alt sum as their conjugate.
%Y A349149 Cf. A000097, A000700, A001700, A027187, A027193, A108711, A236559, A236913, A325534, A344607, A344651.
%K A349149 nonn
%O A349149 0,5
%A A349149 _Gus Wiseman_, Nov 09 2021