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A345195 Number of non-alternating anti-run compositions of n.

%I A345195 #18 Jan 31 2024 15:50:34
%S A345195 0,0,0,0,0,0,2,4,10,23,49,96,192,368,692,1299,2403,4400,8029,14556,
%T A345195 26253,47206,84574,151066,269244,478826,849921,1506309,2665829,
%U A345195 4711971,8319763,14675786,25865400,45552678,80171353,141015313,247905305,435614270,765132824
%N A345195 Number of non-alternating anti-run compositions of n.
%C A345195 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).
%C A345195 An anti-run (separation or Carlitz composition) is a sequence with no adjacent equal parts.
%H A345195 Andrew Howroyd, <a href="/A345195/b345195.txt">Table of n, a(n) for n = 0..1000</a>
%F A345195 a(n) = A003242(n) - A025047(n).
%e A345195 The a(9) = 23 anti-runs:
%e A345195   (1,2,6)  (1,2,4,2)  (1,2,1,2,3)
%e A345195   (1,3,5)  (1,2,5,1)  (1,2,3,1,2)
%e A345195   (2,3,4)  (1,3,4,1)  (1,2,3,2,1)
%e A345195   (4,3,2)  (1,4,3,1)  (1,3,2,1,2)
%e A345195   (5,3,1)  (1,5,2,1)  (2,1,2,3,1)
%e A345195   (6,2,1)  (2,1,2,4)  (2,1,3,2,1)
%e A345195            (2,4,2,1)  (3,2,1,2,1)
%e A345195            (3,1,2,3)
%e A345195            (3,2,1,3)
%e A345195            (4,2,1,2)
%t A345195 wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
%t A345195 sepQ[y_]:=!MatchQ[y,{___,x_,x_,___}];
%t A345195 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], sepQ[#]&&!wigQ[#]&]],{n,0,15}]
%Y A345195 Non-anti-run compositions are counted by A261983.
%Y A345195 A version counting partitions is A345166, ranked by A345173.
%Y A345195 These compositions are ranked by A345169.
%Y A345195 Non-alternating compositions are counted by A345192, ranked by A345168.
%Y A345195 A001250 counts alternating permutations, complement A348615.
%Y A345195 A003242 counts anti-run compositions, ranked by A333489.
%Y A345195 A011782 counts compositions.
%Y A345195 A025047 counts alternating or wiggly compositions, ranked by A345167.
%Y A345195 A032020 counts strict compositions.
%Y A345195 A106356 counts compositions by number of maximal anti-runs.
%Y A345195 A325534 counts separable partitions, ranked by A335433.
%Y A345195 A325535 counts inseparable partitions, ranked by A335448.
%Y A345195 A345164 counts alternating permutations of prime indices, w/ twins A344606.
%Y A345195 A345165 counts partitions w/o an alternating permutation, ranked by A345171.
%Y A345195 A345170 counts partitions w/ an alternating permutation, ranked by A345172.
%Y A345195 A345194 counts alternating patterns (with twins: A344605).
%Y A345195 Cf. A005649, A008965, A114901, A178470, A333755, A344604, A344614, A344654, A344740, A345162, A345163, A348380, A348612, A348613.
%K A345195 nonn
%O A345195 0,7
%A A345195 _Gus Wiseman_, Jun 17 2021
%E A345195 a(21) onwards from _Andrew Howroyd_, Jan 31 2024