A344611 Number of integer partitions of 2n with reverse-alternating sum >= 0.

1, 2, 4, 8, 15, 27, 48, 81, 135, 220, 352, 553, 859, 1313, 1986, 2969, 4394, 6439, 9357, 13479, 19273, 27353, 38558, 53998, 75168, 104022, 143172, 196021, 267051, 362086, 488733, 656802, 879026, 1171747, 1555997, 2058663, 2714133, 3566122, 4670256, 6096924, 7935184

Offset: 0

Author

Gus Wiseman, May 30 2021

Keywords

• nonn

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.

Also the number of reversed integer partitions of 2n with alternating sum >= 0.

The reverse-alternating sum of a partition is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts. So a(n) is the number of partitions of 2n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of partitions of 2n whose parts are all even or whose greatest part is odd.

Examples

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)   (4)     (6)       (8)
      (11)  (22)    (33)      (44)
            (211)   (222)     (332)
            (1111)  (321)     (422)
                    (411)     (431)
                    (2211)    (521)
                    (21111)   (611)
                    (111111)  (2222)
                              (3311)
                              (22211)
                              (32111)
                              (41111)
                              (221111)
                              (2111111)
                              (11111111)
		

Crossrefs

The non-reversed version is A058696 (partitions of 2n).

The ordered version appears to be A114121.

Odd bisection of A344607.

Row sums of A344610.

The strict case is A344650.

A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.

A000070 counts partitions with alternating sum 1.

A000097 counts partitions with alternating sum 2.

A103919 counts partitions by sum and alternating sum.

A120452 counts partitions of 2n with reverse-alternating sum 2.

A316524 is the alternating sum of the prime indices of n (reverse: A344616).

A325534/A325535 count separable/inseparable partitions.

A344612 counts partitions by sum and rev-alt sum (strict: A344739).

A344618 gives reverse-alternating sums of standard compositions.

A344741 counts partitions of 2n with reverse-alternating sum -2.

Cf. A001250, A027187, A028260, A116406, A119899, A124754, A152146, A239829, A344608, A344609, A344649, A344651, A344654.

Programs

Mathematica

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],sats[#]>=0&]],{n,0,30,2}]

Formula

Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...

Extensions

More terms from Bert Dobbelaere, Jun 12 2021