A349061 Number of 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, 1, 2, 4, 8, 13, 21, 32, 48, 67, 99, 133, 185, 245, 333, 432, 574, 732, 957, 1208, 1554, 1941, 2468, 3060, 3844, 4731, 5893, 7204, 8898, 10816, 13268, 16043, 19546, 23523, 28497, 34150, 41147, 49106, 58892, 70020, 83597, 99047, 117778, 139087

Offset: 0

Author

Gus Wiseman, Dec 06 2021

Keywords

• nonn

Comments

Also the number of non-weakly alternating 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.

Examples

			The a(6) = 1 through a(10) = 13 partitions:
  (321)  (421)   (431)    (432)     (532)
         (3211)  (521)    (531)     (541)
                 (4211)   (621)     (631)
                 (32111)  (3321)    (721)
                          (4311)    (4321)
                          (5211)    (5311)
                          (42111)   (6211)
                          (321111)  (32221)
                                    (33211)
                                    (43111)
                                    (52111)
                                    (421111)
                                    (3211111)
		

Crossrefs

The strong version for compositions is A345192, ranked by A345168.

The version for compositions is A349053, ranked by A349057.

The complement is counted by A349060.

These partitions are ranked by A349794.

The non-strict case is A349796, complement A349795.

The strong case is A349801.

A000041 counts integer partitions.

A001250 counts alternating permutations, complement A348615.

A003242 counts Carlitz (anti-run) compositions.

A025047 counts alternating compositions, ranked by A345167.

A025048 and A025049 count directed alternating compositions.

A096441 counts weakly alternating 0-appended partitions.

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

A349052 counts weakly alternating compositions.

A349056 counts weakly alternating permutations of prime indices.

A349798 counts weakly but not strongly alternating perms of prime indices.

Cf. A002865, A027383, A117298, A117989, A129852, A129853, A274230, A333213, A344615, A345165, A349059.

Programs

Mathematica

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