A349796 Number of non-strict integer partitions of n with at least one part of odd multiplicity that is not the first or last.

0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 8, 15, 23, 37, 52, 80, 109, 156, 208, 289, 378, 509, 654, 865, 1098, 1425, 1789, 2290, 2852, 3603, 4450, 5569, 6830, 8467, 10321, 12701, 15393, 18805, 22678, 27535, 33057, 39908, 47701, 57304, 68226, 81572, 96766, 115212, 136201

Offset: 0

Author

Gus Wiseman, Dec 25 2021

Keywords

• nonn

Comments

Also the number of non-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 involves the somewhat degenerate case where no strict increases are allowed.

Examples

			The a(7) = 1 through a(11) = 15 partitions:
  (3211)  (4211)   (3321)    (5311)     (4322)
          (32111)  (4311)    (6211)     (4421)
                   (5211)    (32221)    (5411)
                   (42111)   (33211)    (6311)
                   (321111)  (43111)    (7211)
                             (52111)    (42221)
                             (421111)   (43211)
                             (3211111)  (53111)
                                        (62111)
                                        (322211)
                                        (332111)
                                        (431111)
                                        (521111)
                                        (4211111)
                                        (32111111)
		

Crossrefs

Counting all non-strict partitions gives A047967.

Signatures of this type are counted by A274230, complement A027383.

The strict instead of non-strict version is A347548, ranked by A350352.

The version for compositions allowing strict is A349053, ranked by A349057.

Allowing strict partitions gives A349061, complement A349060.

The complement in non-strict partitions is A349795.

These partitions are ranked by A350140 = A349794 \ A005117.

A000041 = integer partitions, strict A000009.

A001250 = alternating permutations, complement A348615.

A003242 = Carlitz (anti-run) compositions.

A025047 = alternating compositions, ranked by A345167.

A025048/A025049 = directed alternating compositions.

A096441 = weakly alternating 0-appended partitions.

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

A349052 = weakly alternating compositions.

A349056 = weakly alternating permutations of prime indices.

A349798 = weakly but not strongly alternating permutations of prime indices.

Cf. A000111, A002865, A117298, A117989, A129852, A129853, A345165, A345192, A349054, A349059, A349801.

Programs

Mathematica

whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[IntegerPartitions[n],!whkQ[#]&&!whkQ[-#]&&!UnsameQ@@#&]],{n,0,30}]

Formula

a(n) = A349061(n) - A347548(n).