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

0,4

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).

Links

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

Example

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)

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}]

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.

Sequence in context: A014853 A131627 A325712 * A237822 A325713 A325714

Adjacent sequences: A345167 A345168 A345169 * A345171 A345172 A345173

Keyword

nonn

Author

Gus Wiseman, Jun 13 2021

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

Status

approved