A349795 Number of non-strict integer partitions of n that are constant or whose part multiplicities, except possibly the first and last, are all even.

0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 39, 46, 61, 69, 90, 103, 131, 147, 185, 207, 259, 286, 355, 391, 482, 528, 644, 706, 858, 933, 1129, 1228, 1477, 1597, 1916, 2072, 2473, 2668, 3168, 3415, 4047, 4347, 5133, 5514, 6488, 6952, 8162, 8738, 10226, 10936

Offset: 0

Author

Gus Wiseman, Dec 06 2021

Keywords

• nonn

Comments

Also the number of weakly alternating non-strict integer partitions of n, where we define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either. This sequence looks at the somewhat degenerate case where no strict increases are allowed. Equivalently, these are partitions that are weakly alternating but not strongly alternating.

Examples

			The a(2) = 1 through a(8) = 14 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)      (44)
               (211)   (311)    (222)     (331)      (332)
               (1111)  (2111)   (411)     (511)      (422)
                       (11111)  (2211)    (2221)     (611)
                                (3111)    (4111)     (2222)
                                (21111)   (22111)    (3221)
                                (111111)  (31111)    (3311)
                                          (211111)   (5111)
                                          (1111111)  (22211)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
		

Crossrefs

This is the restriction of A349060 to non-strict partitions.

The complement in non-strict partitions is A349796.

Permutations of prime factors of this type are counted by A349798.

The ordered version (compositions) is A349800, ranked by A349799.

These partitions are ranked by A350137.

A000041 counts integer partitions, non-strict A047967.

A001250 counts alternating permutations, complement A348615.

A025047 counts alternating compositions, also A025048 and A025049.

A096441 counts weakly alternating 0-appended partitions.

A345170 counts partitions w/ an alternating permutation, ranked by A345172.

A349053 counts non-weakly alternating compositions, complement A349052.

A349061 counts non-weakly alternating partitions, ranked by A349794.

A349801 counts non-alternating partitions.

Cf. A000070, A003242, A027383, A065033, A117298, A129852, A274230, A344614, A344615, A345165, A345167, A347548, A349056, A349057.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],!UnsameQ@@#&&(SameQ@@#||And@@EvenQ/@Take[Length/@Split[#],{2,-2}])&]],{n,0,30}]

Formula

a(n > 0) = A349060(n) - A065033(n) = A349060(n) - floor(n/2).

a(n) = A047967(n) - A349796(n).