A349801 Number of integer partitions of n into three or more parts or into two equal parts.

0, 0, 1, 1, 3, 4, 8, 11, 18, 25, 37, 50, 71, 94, 128, 168, 223, 288, 376, 480, 617, 781, 991, 1243, 1563, 1945, 2423, 2996, 3704, 4550, 5589, 6826, 8333, 10126, 12293, 14865, 17959, 21618, 25996, 31165, 37318, 44562, 53153, 63239, 75153, 89111, 105535, 124730

Offset

0,5

Comments

This sequence arose as the following degenerate case. If we define a sequence to be alternating if it is alternately strictly increasing and strictly decreasing, starting with either, then a(n) is the number of non-alternating integer partitions of n. Under this interpretation:

- The non-strict case is A047967, weak A349796, weak complement A349795.

- The complement is counted by A065033(n) = ceiling(n/2) for n > 0.

- These partitions are ranked by A289553 \ {1}, complement A167171 \/ {1}.

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

- The weak version for compositions is A349053, ranked by A349057.

- The weak version is A349061, complement A349060, ranked by A349794.

Links

Table of n, a(n) for n=0..47.

Formula

a(1) = 0; a(n > 0) = A000041(n) - ceiling(n/2).

Example

The a(2) = 1 through a(7) = 11 partitions:
  (11)  (111)  (22)    (221)    (33)      (322)
               (211)   (311)    (222)     (331)
               (1111)  (2111)   (321)     (421)
                       (11111)  (411)     (511)
                                (2211)    (2221)
                                (3111)    (3211)
                                (21111)   (4111)
                                (111111)  (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], MatchQ[#, {x_, x_}|{_, _, __}]&]], {n, 0, 10}]

Crossrefs

A000041 counts partitions, ordered A011782.

A001250 counts alternating permutations, complement A348615.

A004250 counts partitions into three or more parts, strict A347548.

A025047/A025048/A025049 count alternating compositions, ranked by A345167.

A096441 counts weakly alternating 0-appended partitions.

A345165 counts partitions w/ no alternating permutation, complement A345170.

Cf. A000070, A001700, A002865, A117298, A117989, A102726, A128761, A345162, A345163, A345166, A349798.

Sequence in context: A208971 A001994 A212544 * A320787 A183151 A288566

Adjacent sequences: A349798 A349799 A349800 * A349802 A349803 A349804

Keyword

nonn

Author

Gus Wiseman, Dec 23 2021

Status

approved