A344608 Number of integer partitions of n with reverse-alternating sum < 0.

0, 0, 0, 1, 1, 3, 3, 7, 7, 14, 15, 27, 29, 49, 54, 86, 96, 146, 165, 242, 275, 392, 449, 623, 716, 973, 1123, 1498, 1732, 2274, 2635, 3411, 3955, 5059, 5871, 7427, 8620, 10801, 12536, 15572, 18065, 22267, 25821, 31602, 36617, 44533, 51560, 62338, 72105, 86716

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 of integer partitions of n with alternating sum < 0.

No integer partitions have alternating sum < 0, so the non-reversed version is all zeros.

Is this sequence weakly increasing? Note: a(2n + 2) = A236914(n), a(2n) = A344743(n).

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 of even length whose conjugate parts are not all odd. Partitions of the latter type are counted by A086543. By conjugation, a(n) is also the number of integer partitions of n of even maximum whose parts are not all odd.

Examples

			The a(3) = 1 through a(9) = 14 partitions:
  (21)  (31)  (32)    (42)    (43)      (53)      (54)
              (41)    (51)    (52)      (62)      (63)
              (2111)  (3111)  (61)      (71)      (72)
                              (2221)    (3221)    (81)
                              (3211)    (4211)    (3222)
                              (4111)    (5111)    (3321)
                              (211111)  (311111)  (4221)
                                                  (4311)
                                                  (5211)
                                                  (6111)
                                                  (222111)
                                                  (321111)
                                                  (411111)
                                                  (21111111)
		

Crossrefs

The opposite version (rev-alt sum > 0) is A027193, ranked by A026424.

The strict case (for n > 2) is A067659 (odd bisection: A344650).

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

The bisections are A236914 (odd) and A344743 (even).

The ordered version appears to be A294175 (even bisection: A008549).

The complement is counted by A344607 (even bisection: A344611).

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

A027187 counts partitions with alternating sum <= 0, ranked by A028260.

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

A120452 counts partitions with rev-alternating sum 2 (negative: A344741).

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

A325534/A325535 count separable/inseparable partitions.

A344604 counts wiggly compositions with twins.

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

A344618 gives reverse-alternating sums of standard compositions.

Cf. A000070, A000097, A000346, A003242, A006330, A071322, A239829, A239830, A344649, A344651, A344654/A344740, A344739.

Programs

Mathematica

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