cp's OEIS Frontend

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.

%I A337507 #10 Sep 17 2020 20:35:06
%S A337507 0,0,1,4,24,176,1540,15672,181916,2372512,34348932,546674120,
%T A337507 9486840748,178285201008,3607174453844,78177409231768,
%U A337507 1806934004612220,44367502983673664,1153334584544496676,31643148872573831016
%N 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.
%C A337507 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 A337507 a(n > 0) = (n - 1)*A005649(n - 2).
%e A337507 The a(4) = 24 sequences:
%e A337507   (2,1,2,2)  (2,1,3,3)  (3,1,2,2)
%e A337507   (2,2,1,2)  (2,3,3,1)  (3,2,2,1)
%e A337507   (1,2,2,1)  (3,3,1,2)  (1,1,2,3)
%e A337507   (2,1,1,2)  (3,3,2,1)  (1,1,3,2)
%e A337507   (1,1,2,1)  (1,2,2,3)  (2,1,1,3)
%e A337507   (1,2,1,1)  (1,3,2,2)  (2,3,1,1)
%e A337507   (1,2,3,3)  (2,2,1,3)  (3,1,1,2)
%e A337507   (1,3,3,2)  (2,2,3,1)  (3,2,1,1)
%t A337507 kv=2;
%t A337507 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A337507 Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==kv&]],{n,0,6}]
%Y A337507 A002133 is the version for runs in partitions.
%Y A337507 A106357 is the version for compositions.
%Y A337507 A337506 has this as column k = 2.
%Y A337507 A000670 counts patterns.
%Y A337507 A005649 counts anti-run patterns.
%Y A337507 A003242 counts anti-run compositions.
%Y A337507 A106356 counts compositions by number of maximal anti-runs.
%Y A337507 A124762 counts adjacent equal terms in standard compositions.
%Y A337507 A124767 counts maximal runs in standard compositions.
%Y A337507 A238130/A238279/A333755 count maximal runs in compositions.
%Y A337507 A333381 counts maximal anti-runs in standard compositions.
%Y A337507 A333382 counts adjacent unequal terms in standard compositions.
%Y A337507 A333489 ranks anti-run compositions.
%Y A337507 A333769 gives maximal run lengths in standard compositions.
%Y A337507 A337565 gives maximal anti-run lengths in standard compositions.
%Y A337507 Cf. A019472, A052841, A060223, A106351, A269134, A335461, A337505, A337564.
%K A337507 nonn
%O A337507 0,4
%A A337507 _Gus Wiseman_, Sep 06 2020