cp's OEIS Frontend

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

%I A349053 #16 Jan 31 2024 15:52:51
%S A349053 0,0,0,0,0,0,4,12,37,95,232,533,1198,2613,5619,11915,25011,52064,
%T A349053 107694,221558,453850,926309,1884942,3825968,7749312,15667596,
%U A349053 31628516,63766109,128415848,258365323,519392582,1043405306,2094829709,4203577778,8431313237,16904555958
%N A349053 Number of non-weakly alternating integer compositions of n.
%C A349053 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.
%H A349053 Andrew Howroyd, <a href="/A349053/b349053.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..55 from Martin Ehrenstein)
%H A349053 Wikipedia, <a href="https://en.wikipedia.org/wiki/Alternating_permutation">Alternating permutation</a>
%F A349053 a(n) = A011782(n) - A349052(n).
%e A349053 The a(6) = 12 compositions:
%e A349053   (1,1,2,2,1)  (1,1,2,3)  (1,2,4)
%e A349053   (1,2,1,1,2)  (1,2,3,1)  (4,2,1)
%e A349053   (1,2,2,1,1)  (1,3,2,1)
%e A349053   (2,1,1,2,1)  (2,1,1,3)
%e A349053                (3,1,1,2)
%e A349053                (3,2,1,1)
%t A349053 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}];
%t A349053 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!wwkQ[#]&]],{n,0,10}]
%Y A349053 Complementary directed versions are A129852/A129853, strong A025048/A025049.
%Y A349053 The strong version is A345192.
%Y A349053 The complement is counted by A349052.
%Y A349053 These compositions are ranked by A349057, strong A345168.
%Y A349053 The complementary version for patterns is A349058, strong A345194.
%Y A349053 The complementary multiplicative version is A349059, strong A348610.
%Y A349053 An unordered version (partitions) is A349061, complement A349060.
%Y A349053 The version for ordered prime factorizations is A349797, complement A349056.
%Y A349053 The version for patterns is A350138, strong A350252.
%Y A349053 The version for ordered factorizations is A350139.
%Y A349053 A001250 counts alternating permutations, complement A348615.
%Y A349053 A001700 counts compositions of 2n with alternating sum 0.
%Y A349053 A003242 counts Carlitz (anti-run) compositions.
%Y A349053 A011782 counts compositions, unordered A000041.
%Y A349053 A025047 counts alternating compositions, ranked by A345167.
%Y A349053 A106356 counts compositions by number of maximal anti-runs.
%Y A349053 A344604 counts alternating compositions with twins.
%Y A349053 A345164 counts alternating ordered prime factorizations.
%Y A349053 A349054 counts strict alternating compositions.
%Y A349053 Cf. A102726, A114901, A128761, A261983, A333213, A333755, A344614, A344615, A345165, A345170, A345195, A349799, A349800, A350251, A350252.
%K A349053 nonn
%O A349053 0,7
%A A349053 _Gus Wiseman_, Dec 16 2021
%E A349053 a(21)-a(35) from _Martin Ehrenstein_, Jan 08 2022