%I #10 Sep 17 2020 20:35:06

%S 0,0,1,4,24,176,1540,15672,181916,2372512,34348932,546674120,

%T 9486840748,178285201008,3607174453844,78177409231768,

%U 1806934004612220,44367502983673664,1153334584544496676,31643148872573831016

%N Number of length-n sequences covering an initial interval of positive integers with exactly two maximal anti-runs, or with one pair of adjacent equal parts.

%C An anti-run is a sequence with no adjacent equal parts. For example, the maximal anti-runs in (3,1,1,2,2,2,1) are ((3,1),(1,2),(2),(2,1)). In general, there is one more maximal anti-run than the number of pairs of adjacent equal parts.

%F a(n > 0) = (n - 1)*A005649(n - 2).

%e The a(4) = 24 sequences:

%e (2,1,2,2) (2,1,3,3) (3,1,2,2)

%e (2,2,1,2) (2,3,3,1) (3,2,2,1)

%e (1,2,2,1) (3,3,1,2) (1,1,2,3)

%e (2,1,1,2) (3,3,2,1) (1,1,3,2)

%e (1,1,2,1) (1,2,2,3) (2,1,1,3)

%e (1,2,1,1) (1,3,2,2) (2,3,1,1)

%e (1,2,3,3) (2,2,1,3) (3,1,1,2)

%e (1,3,3,2) (2,2,3,1) (3,2,1,1)

%t kv=2;

%t allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];

%t Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==kv&]],{n,0,6}]

%Y A002133 is the version for runs in partitions.

%Y A106357 is the version for compositions.

%Y A337506 has this as column k = 2.

%Y A000670 counts patterns.

%Y A005649 counts anti-run patterns.

%Y A003242 counts anti-run compositions.

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

%Y A124762 counts adjacent equal terms in standard compositions.

%Y A124767 counts maximal runs in standard compositions.

%Y A238130/A238279/A333755 count maximal runs in compositions.

%Y A333381 counts maximal anti-runs in standard compositions.

%Y A333382 counts adjacent unequal terms in standard compositions.

%Y A333489 ranks anti-run compositions.

%Y A333769 gives maximal run lengths in standard compositions.

%Y A337565 gives maximal anti-run lengths in standard compositions.

%Y Cf. A019472, A052841, A060223, A106351, A269134, A335461, A337505, A337564.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 06 2020