login
Number of even-length integer partitions of n with at most one odd part in the conjugate partition.
2

%I #8 Nov 12 2021 22:31:54

%S 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 101,272,135,373,176,508,231,684,297,915,385,1212,490,1597,627,2087,

%U 792,2714,1002,3506,1255,4508,1575,5763,1958,7338,2436,9296,3010,11732

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

%C 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 a(2n) = A000041(n).

%F a(2n+1) = A000070(n-1).

%e The a(2) = 1 through a(9) = 7 partitions:

%e 11 21 22 32 33 43 44 54

%e 1111 2111 2211 2221 2222 3222

%e 111111 3211 3311 3321

%e 211111 221111 4311

%e 11111111 222111

%e 321111

%e 21111111

%t conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];

%t Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[conj[#],_?OddQ]<=1&]],{n,0,30}]

%Y The case of 0 odd conjugate parts is A000041 up to 0's, ranked by A000290.

%Y The case of 1 odd conjugate part is A000070 up to 0's.

%Y Even bisection of A100824, ranked by A349150.

%Y Ranked by A349151 /\ A028260.

%Y A045931 counts partitions with as many even as odd parts, ranked by A325698.

%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).

%Y A122111 is a representation of partition conjugation.

%Y A277103 counts partitions with the same alternating sum as their conjugate.

%Y A277579 counts partitions with as many even parts as odd conjugate parts.

%Y A325039 counts partitions with the same product as their conjugate.

%Y A344610 counts partitions by sum and positive reverse-alternating sum.

%Y A345196 counts partitions with the same rev-alt sum as their conjugate.

%Y Cf. A000097, A000700, A001700, A027187, A027193, A108711, A236559, A236913, A325534, A344607, A344651.

%K nonn

%O 0,5

%A _Gus Wiseman_, Nov 09 2021