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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.

Cf. A019472, A052841, A060223, A106351, A269134, A325535, A335461, A337505, A337564.

A106356 counts compositions by number of maximal anti-runs.

Cf. A019472, A052841, A060223, A106351, A106356, A269134, A325535, A335461, A337505, A337564.

Number of length-n sequences of length n covering an initial interval of positive integers with 2 two maximal anti-runs, or with 1 one pair of adjacent equal parts.

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.

nonn,more,changed

0, 0, 1, 4, 24, 176, 1540, 15672, 181916, 2372512, 34348932, 546674120, 9486840748, 178285201008, 3607174453844, 78177409231768, 1806934004612220, 44367502983673664, 1153334584544496676, 31643148872573831016

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

allocated for Gus WisemanNumber of sequences of length n covering an initial interval of positive integers with 2 maximal anti-runs, or with 1 pair of adjacent equal parts.

0, 0, 1, 4, 24, 176, 1540, 15672, 181916, 2372512

0,4

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.

The a(4) = 24 sequences:

(2,1,2,2) (2,1,3,3) (3,1,2,2)

(2,2,1,2) (2,3,3,1) (3,2,2,1)

(1,2,2,1) (3,3,1,2) (1,1,2,3)

(2,1,1,2) (3,3,2,1) (1,1,3,2)

(1,1,2,1) (1,2,2,3) (2,1,1,3)

(1,2,1,1) (1,3,2,2) (2,3,1,1)

(1,2,3,3) (2,2,1,3) (3,1,1,2)

(1,3,3,2) (2,2,3,1) (3,2,1,1)

kv=2;

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

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

A002133 is the version for runs in partitions.

A106357 is the version for compositions.

A337506 has this as column k = 2.

A000670 counts patterns.

A005649 counts anti-run patterns.

A003242 counts anti-run compositions.

A124762 counts adjacent equal terms in standard compositions.

A124767 counts maximal runs in standard compositions.

A238130/A238279/A333755 count maximal runs in compositions.

A333381 counts maximal anti-runs in standard compositions.

A333382 counts adjacent unequal terms in standard compositions.

A333489 ranks anti-run compositions.

A333769 gives maximal run lengths in standard compositions.

A337565 gives maximal anti-run lengths in standard compositions.

Cf. A019472, A052841, A060223, A106351, A106356, A269134, A325535, A335461, A337505, A337564.

allocated

nonn,more

Gus Wiseman, Sep 06 2020

approved

editing

allocated for Gus Wiseman

allocated

approved