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

0,4

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.

Links

Table of n, a(n) for n=0..40.

Formula

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

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

Example

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)

Mathematica

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

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.

Sequence in context: A192960 A182717 A122768 * A321309 A023608 A218189

Adjacent sequences: A344740 A344741 A344742 * A344744 A344745 A344746

Keyword

nonn

Author

Gus Wiseman, Jun 09 2021

Extensions

More terms from Bert Dobbelaere, Jun 12 2021

Status

approved