A025048 Number of up/down (initially ascending) compositions of n.

1, 1, 1, 2, 3, 4, 7, 11, 16, 26, 41, 64, 100, 158, 247, 389, 612, 960, 1509, 2372, 3727, 5858, 9207, 14468, 22738, 35737, 56164, 88268, 138726, 218024, 342652, 538524, 846358, 1330160, 2090522, 3285526, 5163632, 8115323, 12754288, 20045027, 31503382

Offset

0,4

Comments

Original name was: Ascending wiggly sums: number of sums adding to n in which terms alternately increase and decrease.

A composition is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2). - Gus Wiseman, Jan 15 2022

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Wikipedia, Alternating permutation

Formula

a(n) = 1+A025047(n)-A025049(n) = sum_k[A059882(n, k)]. - Henry Bottomley, Feb 05 2001

a(n) ~ c * d^n, where d = 1.571630806607064114100138865739690782401305155950789062725011227781640624..., c = 0.4408955566119650057730070154620695491718230084159159991449729825619... . - Vaclav Kotesovec, Sep 12 2014

Example

From Gus Wiseman, Jan 15 2022: (Start)
The a(1) = 1 through a(7) = 11 up/down compositions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)
            (1,2)  (1,3)    (1,4)    (1,5)      (1,6)
                   (1,2,1)  (2,3)    (2,4)      (2,5)
                            (1,3,1)  (1,3,2)    (3,4)
                                     (1,4,1)    (1,4,2)
                                     (2,3,1)    (1,5,1)
                                     (1,2,1,2)  (2,3,2)
                                                (2,4,1)
                                                (1,2,1,3)
                                                (1,3,1,2)
                                                (1,2,1,2,1)
(End)

Mathematica

updoQ[y_]:=And@@Table[If[EvenQ[m], y[[m]]>y[[m+1]], y[[m]]<y[[m+1]]], {m, 1, Length[y]-1}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], updoQ]], {n, 0, 15}] (* Gus Wiseman, Jan 15 2022 *)

Crossrefs

The case of permutations is A000111.

The undirected version is A025047, ranked by A345167.

The down/up version is A025049, ranked by A350356.

The strict case is A129838, undirected A349054.

The weak version is A129852, down/up A129853.

The version for patterns is A350354.

These compositions are ranked by A350355.

A001250 counts alternating permutations, complement A348615.

A003242 counts Carlitz compositions, complement A261983.

A011782 counts compositions, unordered A000041.

A325534 counts separable partitions, complement A325535.

A345192 counts non-alternating compositions, ranked by A345168.

A345194 counts alternating patterns, complement A350252.

A349052 counts weakly alternating compositions, complement A349053.

Cf. A008965, A049774, A128761, A344604, A344605, A344614, A344615, A345195, A349057, A349800.

Sequence in context: A113435 A367667 A222022 * A017987 A337495 A222023

Adjacent sequences: A025045 A025046 A025047 * A025049 A025050 A025051

Keyword

nonn

Author

David W. Wilson

Extensions

Name and offset changed by Gus Wiseman, Jan 15 2022

Status

approved