A345192 Number of non-alternating compositions of n.

0, 0, 1, 1, 4, 9, 20, 45, 99, 208, 437, 906, 1862, 3803, 7732, 15659, 31629, 63747, 128258, 257722, 517339, 1037652, 2079984, 4167325, 8346204, 16710572, 33449695, 66944254, 133959021, 268028868, 536231903, 1072737537, 2145905285, 4292486690, 8586035993, 17173742032, 34350108745, 68704342523, 137415168084

Offset

0,5

Comments

First differs from A261983 at a(6) = 20, A261983(6) = 18.

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 does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2).

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = A011782(n) - A025047(n).

Example

The a(2) = 1 through a(6) = 20 compositions:
  (11)  (111)  (22)    (113)    (33)
               (112)   (122)    (114)
               (211)   (221)    (123)
               (1111)  (311)    (222)
                       (1112)   (321)
                       (1121)   (411)
                       (1211)   (1113)
                       (2111)   (1122)
                       (11111)  (1131)
                                (1221)
                                (1311)
                                (2112)
                                (2211)
                                (3111)
                                (11112)
                                (11121)
                                (11211)
                                (12111)
                                (21111)
                                (111111)

Mathematica

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

Crossrefs

The complement is counted by A025047 (ascend: A025048, descend: A025049).

Dominates A261983 (non-anti-run compositions), ranked by A348612.

These compositions are ranked by A345168, complement A345167.

The case without twins is A348377.

The version for factorizations is A348613.

A001250 counts alternating permutations, complement A348615.

A003242 counts anti-run compositions.

A011782 counts compositions.

A032020 counts strict compositions.

A106356 counts compositions by number of maximal anti-runs.

A114901 counts compositions where each part is adjacent to an equal part.

A274174 counts compositions with equal parts contiguous.

A325534 counts separable partitions, ranked by A335433.

A325535 counts inseparable partitions, ranked by A335448.

A344604 counts alternating compositions with twins.

A344605 counts alternating patterns with twins.

A344654 counts non-twin partitions with no alternating permutation.

A345162 counts normal partitions with no alternating permutation.

A345164 counts alternating permutations of prime indices.

A345170 counts partitions w/ alternating permutation, ranked by A345172.

A345165 counts partitions w/o alternating permutation, ranked by A345171.

Patterns:

- A128761 avoiding (1,2,3) adjacent.

- A344614 avoiding (1,2,3) and (3,2,1) adjacent.

- A344615 weakly avoiding (1,2,3) adjacent.

Cf. A000070, A008965, A178470, A238279, A333755, A335126, A344606, A344653, A344740, A345163, A345166, A345169, A345173, A348380.

Sequence in context: A019493 A019492 A020708 * A109110 A366726 A108870

Adjacent sequences: A345189 A345190 A345191 * A345193 A345194 A345195

Keyword

nonn

Author

Gus Wiseman, Jun 17 2021

Status

approved