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A349149
Number of even-length integer partitions of n with at most one odd part in the conjugate partition.
2
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
0,5
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.
FORMULA
a(2n) = A000041(n).
a(2n+1) = A000070(n-1).
EXAMPLE
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
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}]
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.
Sequence in context: A050322 A325512 A211858 * A029012 A095699 A112192
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 09 2021
STATUS
approved