A349053 Number of non-weakly alternating integer compositions of n.

0, 0, 0, 0, 0, 0, 4, 12, 37, 95, 232, 533, 1198, 2613, 5619, 11915, 25011, 52064, 107694, 221558, 453850, 926309, 1884942, 3825968, 7749312, 15667596, 31628516, 63766109, 128415848, 258365323, 519392582, 1043405306, 2094829709, 4203577778, 8431313237, 16904555958

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. Then a sequence is (strongly) alternating iff it is a weakly alternating anti-run.

Examples

			The a(6) = 12 compositions:
  (1,1,2,2,1)  (1,1,2,3)  (1,2,4)
  (1,2,1,1,2)  (1,2,3,1)  (4,2,1)
  (1,2,2,1,1)  (1,3,2,1)
  (2,1,1,2,1)  (2,1,1,3)
               (3,1,1,2)
               (3,2,1,1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms 0..55 from Martin Ehrenstein)
• Wikipedia, Alternating permutation

Crossrefs

Complementary directed versions are A129852/A129853, strong A025048/A025049.

The strong version is A345192.

The complement is counted by A349052.

These compositions are ranked by A349057, strong A345168.

The complementary version for patterns is A349058, strong A345194.

The complementary multiplicative version is A349059, strong A348610.

An unordered version (partitions) is A349061, complement A349060.

The version for ordered prime factorizations is A349797, complement A349056.

The version for patterns is A350138, strong A350252.

The version for ordered factorizations is A350139.

A001250 counts alternating permutations, complement A348615.

A001700 counts compositions of 2n with alternating sum 0.

A003242 counts Carlitz (anti-run) compositions.

A011782 counts compositions, unordered A000041.

A025047 counts alternating compositions, ranked by A345167.

A106356 counts compositions by number of maximal anti-runs.

A344604 counts alternating compositions with twins.

A345164 counts alternating ordered prime factorizations.

A349054 counts strict alternating compositions.

Cf. A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345170, A345195, A349799, A349800, A350251, A350252.

Programs

Mathematica

wwkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}]||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],!wwkQ[#]&]],{n,0,10}]

Formula

a(n) = A011782(n) - A349052(n).

Extensions

a(21)-a(35) from Martin Ehrenstein, Jan 08 2022