A345165 Number of integer partitions of n without an alternating permutation.

0, 0, 1, 1, 2, 2, 5, 5, 8, 11, 17, 20, 29, 37, 51, 65, 85, 106, 141, 175, 223, 277, 351, 432, 540, 663, 820, 999, 1226, 1489, 1817, 2192, 2654, 3191, 3847, 4603, 5517, 6578, 7853, 9327, 11084, 13120, 15533, 18328, 21621, 25430, 29905, 35071, 41111, 48080, 56206, 65554, 76420, 88918

Offset: 0

Author

Gus Wiseman, Jun 12 2021

Keywords

• nonn

Comments

A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it has the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

Examples

			The a(2) = 1 through a(9) = 11 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)        (333)
               (1111)  (11111)  (222)     (4111)     (2222)      (3222)
                                (3111)    (31111)    (5111)      (6111)
                                (21111)   (211111)   (41111)     (22221)
                                (111111)  (1111111)  (221111)    (51111)
                                                     (311111)    (321111)
                                                     (2111111)   (411111)
                                                     (11111111)  (2211111)
                                                                 (3111111)
                                                                 (21111111)
                                                                 (111111111)
		

Links

• Joseph Likar, Table of n, a(n) for n = 0..1000
• Joseph Likar, Java Implementation using QBinomials

Crossrefs

Excluding twins (x,x) gives A344654, complement A344740.

The normal case is A345162, complement A345163.

The complement is counted by A345170, ranked by A345172.

The Heinz numbers of these partitions are A345171.

The version for factorizations is A348380, complement A348379.

A version for ordered factorizations is A348613, complement A348610.

A000041 counts integer partitions.

A001250 counts alternating permutations, complement A348615.

A003242 counts anti-run compositions.

A005649 counts anti-run patterns.

A025047 counts alternating or wiggly compositions.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts alternating compositions with twins.

A345164 counts alternating permutations of prime indices, w/ twins A344606.

A345192 counts non-alternating compositions, without twins A348377.

Cf. A000070, A025048, A025049, A103919, A335126, A344605, A344607, A344615, A344653, A345166, A345167, A345168, A345169, A347706, A348609.

Programs

Mathematica

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[IntegerPartitions[n],Select[Permutations[#],wigQ]=={}&]],{n,0,15}]

Extensions

a(26) onwards by Joseph Likar, Aug 21 2023