A344743 Number of integer partitions of 2n with reverse-alternating sum < 0.

0, 0, 1, 3, 7, 15, 29, 54, 96, 165, 275, 449, 716, 1123, 1732, 2635, 3955, 5871, 8620, 12536, 18065, 25821, 36617, 51560, 72105, 100204, 138417, 190134, 259772, 353134, 477734, 643354, 862604, 1151773, 1531738, 2029305, 2678650, 3523378, 4618835, 6035240, 7861292

Offset: 0

Author

Gus Wiseman, Jun 09 2021

Keywords

• nonn

Comments

Conjecture: a(n) >= A236914.

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So a(n) is the number of even-length partitions of 2n with at least one odd conjugate part. By conjugation, this is also the number of partitions of 2n with greatest part even and at least one odd part.

The alternating sum of a partition is never < 0, so the non-reverse version is A000004.

Examples

			The a(2) = 1 through a(5) = 15 partitions:
  (31)  (42)    (53)      (64)
        (51)    (62)      (73)
        (3111)  (71)      (82)
                (3221)    (91)
                (4211)    (3331)
                (5111)    (4222)
                (311111)  (4321)
                          (5221)
                          (5311)
                          (6211)
                          (7111)
                          (322111)
                          (421111)
                          (511111)
                          (31111111)
		

Crossrefs

The ordered version (compositions not partitions) appears to be A008549.

The Heinz numbers are A119899 /\ A300061.

Even bisection of A344608.

The complementary partitions of 2n are counted by A344611.

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

A001523 counts unimodal compositions (partial sums: A174439).

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

A120452 counts partitions of 2n with rev-alt sum 2 (negative: A344741).

A124754 gives alternating sums of standard compositions (reverse: A344618).

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

A325534/A325535 count separable/inseparable partitions.

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

Cf. A000070, A000097, A006330, A027187, A239830, A343941, A344604, A344607, A344650, 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

a(n) = A058696(n) - A344611(n).

a(n) = sum of left half of even-indexed rows of A344612.

Extensions

More terms from Bert Dobbelaere, Jun 12 2021