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Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.
50

%I #14 Jun 12 2021 06:05:12

%S 1,1,2,4,8,16,30,58,110,209,397,753,1429,2711,5143,9757,18511,35117,

%T 66621,126389,239781,454897,863010,1637260,3106138,5892821,11179603,

%U 21209446,40237641,76337091,144823431,274752731,521249018,988891100,1876081530,3559220898,6752400377

%N Number of compositions of n with no adjacent triples (..., x, y, z, ...) where x < y < z or x > y > z.

%C These compositions avoid the strict consecutive patterns (1,2,3) and (3,2,1), the weak version being A344604.

%e The a(6) = 30 compositions are:

%e (6) (15) (114) (1113) (11112) (111111)

%e (24) (132) (1122) (11121)

%e (33) (141) (1131) (11211)

%e (42) (213) (1212) (12111)

%e (51) (222) (1221) (21111)

%e (231) (1311)

%e (312) (2112)

%e (411) (2121)

%e (2211)

%e (3111)

%e Missing are: (123), (321).

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MatchQ[#,{___,x_,y_,z_,___}/;x<y<z||x>y>z]&]],{n,0,15}]

%Y A001250 counts alternating permutations.

%Y A005649 counts anti-run patterns.

%Y A025047 counts wiggly compositions (ascend: A025048, descend: A025049).

%Y A106356 counts compositions by number of maximal anti-runs.

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

%Y A325534 counts separable partitions.

%Y A325535 counts inseparable partitions.

%Y A344604 counts wiggly compositions with twins.

%Y A344605 counts wiggly patterns with twins.

%Y A344606 counts wiggly permutations of prime factors with twins.

%Y Counting compositions by patterns:

%Y - A003242 avoiding (1,1) adjacent.

%Y - A011782 no conditions.

%Y - A106351 avoiding (1,1) adjacent by sum and length.

%Y - A128695 avoiding (1,1,1) adjacent.

%Y - A128761 avoiding (1,2,3).

%Y - A232432 avoiding (1,1,1).

%Y - A335456 all patterns.

%Y - A335457 all patterns adjacent.

%Y - A335514 matching (1,2,3).

%Y - A344604 weakly avoiding (1,2,3) and (3,2,1) adjacent.

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

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

%Y Cf. A000041, A006330, A008965, A049774, A056986, A238279/A333755, A333213, A335515, A344612, A344652.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 27 2021

%E More terms from _Bert Dobbelaere_, Jun 12 2021