login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.
6

%I #12 Dec 31 2020 18:34:39

%S 1,2,24,440,10780,329112,12006456,508903824,24559486380,1328964785720,

%T 79670488601704,5240336913228144,375167786246499064,

%U 29038998659140223600,2416268289647552828400,215068032231876851531040,20389611819955706893052460,2051184695261785540782403320

%N Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.

%C An anti-run is a sequence with no adjacent equal parts.

%H Andrew Howroyd, <a href="/A337505/b337505.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = A005649(n)*binomial(2*n-1,n). - _Andrew Howroyd_, Dec 31 2020

%e The a(2) = 24 sequences:

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

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

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

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

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

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

%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[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,3}]

%o (PARI) \\ here b(n) is A005649.

%o b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}

%o a(n) = {b(n)*binomial(2*n-1,n)} \\ _Andrew Howroyd_, Dec 31 2020

%Y A336108 is the version for compositions and runs.

%Y A337504 is the version for compositions.

%Y A337506 has this as main diagonal n = 2*k.

%Y A337564 is the version for runs.

%Y A000670 counts sequences covering an initial interval.

%Y A003242 counts anti-run compositions.

%Y A005649 counts anti-runs covering an initial interval.

%Y A124767 counts maximal runs in standard compositions.

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

%Y A333769 gives run-lengths in standard compositions.

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

%Y Cf. A052841, A106351, A106356, A269134, A325535, A333489, A333627, A333755.

%K nonn

%O 0,2

%A _Gus Wiseman_, Sep 05 2020

%E Terms a(5) and beyond from _Andrew Howroyd_, Dec 31 2020