A347443 Number of integer partitions of n with reverse-alternating product <= 1.

1, 1, 1, 2, 3, 5, 6, 10, 12, 19, 22, 34, 40, 60, 69, 101, 118, 168, 195, 272, 317, 434, 505, 679, 793, 1050, 1224, 1599, 1867, 2409, 2811, 3587, 4186, 5290, 6168, 7724, 9005, 11186, 13026, 16062, 18692, 22894, 26613, 32394, 37619, 45535, 52815, 63593, 73680

Offset: 0

Author

Gus Wiseman, Sep 14 2021

Keywords

• nonn

Comments

Includes all partitions of even length (A027187).

Also the number of integer partitions of n with reverse-alternating sum <= 1.

Also the number of integer partitions of n having either even length (A027187) or having exactly one odd part in the conjugate partition (A100824).

We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.

Examples

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (2111)   (2211)    (331)      (71)
                            (11111)  (3111)    (2221)     (2222)
                                     (111111)  (3211)     (3221)
                                               (4111)     (3311)
                                               (22111)    (4211)
                                               (211111)   (5111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (11111111)
		

Crossrefs

The odd-length case is A035363 (shifted).

The strict case is A067661.

The non-reverse version is counted by A119620, ranked by A347466.

The even bisection is A236913.

The opposite version (>= instead of <=) is A344607.

The case of < 1 instead of <= 1 is A344608.

The multiplicative version (factorizations) is A347438, non-reverse A339846.

Allowing any integer reverse-alternating product gives A347445.

The complement (> 1 instead of <= 1) is counted by A347449.

Ranked by A347465, non-reverse A347450.

A000041 counts partitions.

A027187 counts partitions of even length.

A027193 counts partitions of odd length.

A058622 counts compositions with alternating sum <= 0 (A294175 for < 0).

A100824 counts partitions with alternating sum <= 1.

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

A347461 counts possible alternating products of partitions.

A347462 counts possible reverse-alternating products of partitions.

Cf. A000070, A038548, A086543, A116406, A325534, A325535, A344611, A344654, A344740, A347440, A347442, A347446, A347448.

Programs

Mathematica

altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]<=1&]],{n,0,30}]

Formula

a(n) = A027187(n) + A035363(n-1) for n >= 1. [Corrected by Georg Fischer, Dec 13 2022]

a(n) = A119620(n) + A344608(n).