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

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, 101, 272, 135, 373, 176, 508, 231, 684, 297, 915, 385, 1212, 490, 1597, 627, 2087, 792, 2714, 1002, 3506, 1255, 4508, 1575, 5763, 1958, 7338, 2436, 9296, 3010, 11732

Offset: 0

Author

Gus Wiseman, Nov 09 2021

Keywords

• nonn

Comments

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.

Examples

			The a(2) = 1 through a(9) = 7 partitions:
  11   21   22     32     33       43       44         54
            1111   2111   2211     2221     2222       3222
                          111111   3211     3311       3321
                                   211111   221111     4311
                                            11111111   222111
                                                       321111
                                                       21111111
		

Crossrefs

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

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

Even bisection of A100824, ranked by A349150.

Ranked by A349151 /\ A028260.

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

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

A122111 is a representation of partition conjugation.

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

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

A325039 counts partitions with the same product as their conjugate.

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

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

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

Programs

Mathematica

conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[conj[#],_?OddQ]<=1&]],{n,0,30}]

Formula

a(2n) = A000041(n).

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