A345170 Number of integer partitions of n with an alternating permutation.

1, 1, 1, 2, 3, 5, 6, 10, 14, 19, 25, 36, 48, 64, 84, 111, 146, 191, 244, 315, 404, 515, 651, 823, 1035, 1295, 1616, 2011, 2492, 3076, 3787, 4650, 5695, 6952, 8463, 10280, 12460, 15059, 18162, 21858, 26254, 31463, 37641, 44933, 53554, 63704, 75653, 89683, 106162, 125445, 148020

Offset: 0

Author

Gus Wiseman, Jun 13 2021

Keywords

• nonn

Comments

First differs from A325534 at a(10) = 25, A325534(10) = 26. The first separable partition without an alternating permutation is (3,2,2,2,1).

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

Examples

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)  (3)   (4)    (5)    (6)     (7)      (8)
            (21)  (31)   (32)   (42)    (43)     (53)
                  (211)  (41)   (51)    (52)     (62)
                         (221)  (321)   (61)     (71)
                         (311)  (411)   (322)    (332)
                                (2211)  (331)    (422)
                                        (421)    (431)
                                        (511)    (521)
                                        (3211)   (611)
                                        (22111)  (3221)
                                                 (3311)
                                                 (4211)
                                                 (22211)
                                                 (32111)
		

Links

• Joseph Likar, Table of n, a(n) for n = 0..1000

Crossrefs

Includes all strict partitions A000009.

Including twins (x,x) gives A344740.

The normal case is A345163 (complement: A345162).

The complement is counted by A345165, ranked by A345171.

The Heinz numbers of these partitions are A345172.

The version for factorizations is A348379.

A000041 counts integer partitions.

A001250 counts alternating permutations.

A003242 counts anti-run compositions.

A005649 counts anti-run patterns.

A025047 counts alternating compositions (ascend: A025048, descend: A025049).

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts alternating compositions with twins.

Cf. A000070, A103919, A335126, A344605, A344653, A344654, A344742, A345164, A345166, A345167, A345168, A345195.

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)-a(32) from Robert Price, Jun 23 2021

a(33)-a(48) from Alois P. Heinz, Jun 23 2021

a(49) onwards from Joseph Likar, Sep 05 2023