A337507 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.

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

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..19.

Formula

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

Example

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)

Mathematica

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}]

Crossrefs

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.

A106356 counts compositions by number of maximal anti-runs.

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, A269134, A335461, A337505, A337564.

Sequence in context: A112914 A308543 A007846 * A139702 A213591 A243689

Adjacent sequences: A337504 A337505 A337506 * A337508 A337509 A337510

Keyword

nonn

Author

Gus Wiseman, Sep 06 2020

Status

approved