A344607 Number of integer partitions of n with reverse-alternating sum >= 0.

1, 1, 2, 2, 4, 4, 8, 8, 15, 16, 27, 29, 48, 52, 81, 90, 135, 151, 220, 248, 352, 400, 553, 632, 859, 985, 1313, 1512, 1986, 2291, 2969, 3431, 4394, 5084, 6439, 7456, 9357, 10836, 13479, 15613, 19273, 22316, 27353, 31659, 38558, 44601, 53998, 62416, 75168

Offset: 0

Author

Gus Wiseman, May 29 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 n with alternating sum >= 0.

A formula for the reverse-alternating sum of a partition is: (-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 integer partitions of n whose conjugate parts are all even or whose length is odd. By conjugation, this is also the number of integer partitions of n whose parts are all even or whose greatest part is odd.

All integer partitions have alternating sum >= 0, so the non-reversed version is A000041.

Is this sequence weakly increasing? In particular, is A344611(n) <= A160786(n)?

Examples

			The a(1) = 1 through a(8) = 15 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (221)    (33)      (322)      (44)
                    (211)   (311)    (222)     (331)      (332)
                    (1111)  (11111)  (321)     (421)      (422)
                                     (411)     (511)      (431)
                                     (2211)    (22111)    (521)
                                     (21111)   (31111)    (611)
                                     (111111)  (1111111)  (2222)
                                                          (3311)
                                                          (22211)
                                                          (32111)
                                                          (41111)
                                                          (221111)
                                                          (2111111)
                                                          (11111111)
		

Crossrefs

The non-reversed version is A000041.

The opposite version (rev-alt sum <= 0) is A027187, ranked by A028260.

The strict case for n > 0 is A067659 (even bisection: A344650).

The ordered version appears to be A116406 (even bisection: A114121).

The odd bisection is A160786.

The complement is counted by A344608.

The Heinz numbers of these partitions are A344609 (complement: A119899).

The even bisection is A344611.

A000070 counts partitions with alternating sum 1 (reversed: A000004).

A000097 counts partitions with alternating sum 2 (reversed: A120452).

A035363 counts partitions with alternating sum 0, ranked by A000290.

A103919 counts partitions by sum and alternating sum.

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

A325534/A325535 count separable/inseparable partitions.

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

A344612 counts partitions by sum and reverse-alternating sum.

A344618 gives reverse-alternating sums of standard compositions.

Cf. A006330, A071321, A071322, A124754, A239829, A239830, A344604, A344651, A344654, A344739, A344742.

Programs

Mathematica

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

Formula

a(n) + A344608(n) = A000041(n).

a(2n+1) = A160786(n).