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

0,5

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.

Links

Table of n, a(n) for n=0..51.

Formula

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

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

Example

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)

Mathematica

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

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.

Sequence in context: A098390 A266769 A008763 * A005896 A147953 A163468

Adjacent sequences: A349792 A349793 A349794 * A349796 A349797 A349798

Keyword

nonn

Author

Gus Wiseman, Dec 06 2021

Status

approved