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.
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.
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.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 06 2020
STATUS
approved