A349800 Number of integer compositions of n that are weakly alternating and have at least two adjacent equal parts.

0, 0, 1, 1, 4, 9, 16, 33, 62, 113, 205, 373, 664, 1190, 2113, 3744, 6618, 11683, 20564, 36164, 63489, 111343, 195042, 341357, 596892, 1042976, 1821179, 3178145, 5543173, 9663545, 16839321, 29332231, 51075576, 88908912, 154722756, 269186074, 468221264

Offset: 0

Author

Gus Wiseman, Dec 16 2021

Keywords

• nonn

Comments

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

This sequence counts compositions that are weakly but not strongly alternating; also weakly alternating non-anti-run compositions.

Examples

			The a(2) = 1 through a(6) = 16 compositions:
  (1,1)  (1,1,1)  (2,2)      (1,1,3)      (3,3)
                  (1,1,2)    (1,2,2)      (1,1,4)
                  (2,1,1)    (2,2,1)      (2,2,2)
                  (1,1,1,1)  (3,1,1)      (4,1,1)
                             (1,1,1,2)    (1,1,1,3)
                             (1,1,2,1)    (1,1,2,2)
                             (1,2,1,1)    (1,1,3,1)
                             (2,1,1,1)    (1,3,1,1)
                             (1,1,1,1,1)  (2,2,1,1)
                                          (3,1,1,1)
                                          (1,1,1,1,2)
                                          (1,1,1,2,1)
                                          (1,1,2,1,1)
                                          (1,2,1,1,1)
                                          (2,1,1,1,1)
                                          (1,1,1,1,1,1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Wikipedia, Alternating permutation

Crossrefs

This is the weakly alternating case of A345192, ranked by A345168.

The case of partitions is A349795, ranked by A350137.

The version counting permutations of prime indices is A349798.

These compositions are ranked by A349799.

A001250 = alternating permutations, ranked by A349051, complement A348615.

A003242 = Carlitz (anti-run) compositions, ranked by A333489.

A025047/A025048/A025049 = alternating compositions, ranked by A345167.

A261983 = non-anti-run compositions, ranked by A348612.

A345165 = partitions without an alternating permutation, ranked by A345171.

A345170 = partitions with an alternating permutation, ranked by A345172.

A345173 = non-alternating anti-run partitions, ranked by A345166.

A345195 = non-alternating anti-run compositions, ranked by A345169.

A348377 = non-alternating non-twin compositions.

A349801 = non-alternating partitions, ranked by A289553.

Weakly alternating:

- A349052 = compositions, directed A129852/A129853, complement A349053.

- A349056 = permutations of prime indices, complement A349797.

- A349057 = complement of standard composition numbers (too dense).

- A349058 = patterns, complement A350138.

- A349059 = ordered factorizations, complement A350139.

- A349060 = partitions, complement A349061.

Cf. A008965, A011782, A027383, A096441, A274230, A333213, A344614, A344615, A348382, A348613, A349796, A350140.

Programs

Mathematica

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y] &&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
whkQ[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],(whkQ[#]||whkQ[-#])&&!wigQ[#]&]],{n,0,10}]

Formula

a(n) = A349052(n) - A025047(n). - Andrew Howroyd, Jan 31 2024

Extensions

a(21) onwards from Andrew Howroyd, Jan 31 2024