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A337505 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.

%I A337505 #12 Dec 31 2020 18:34:39
%S A337505 1,2,24,440,10780,329112,12006456,508903824,24559486380,1328964785720,
%T A337505 79670488601704,5240336913228144,375167786246499064,
%U A337505 29038998659140223600,2416268289647552828400,215068032231876851531040,20389611819955706893052460,2051184695261785540782403320
%N A337505 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.
%C A337505 An anti-run is a sequence with no adjacent equal parts.
%H A337505 Andrew Howroyd, <a href="/A337505/b337505.txt">Table of n, a(n) for n = 0..200</a>
%F A337505 a(n) = A005649(n)*binomial(2*n-1,n). - _Andrew Howroyd_, Dec 31 2020
%e A337505 The a(2) = 24 sequences:
%e A337505   (2,1,2,2)  (1,2,3,3)  (1,2,2,3)  (1,1,2,3)
%e A337505   (2,2,1,2)  (1,3,3,2)  (1,3,2,2)  (1,1,3,2)
%e A337505   (1,2,2,1)  (2,1,3,3)  (2,2,1,3)  (2,1,1,3)
%e A337505   (2,1,1,2)  (2,3,3,1)  (2,2,3,1)  (2,3,1,1)
%e A337505   (1,1,2,1)  (3,3,1,2)  (3,1,2,2)  (3,1,1,2)
%e A337505   (1,2,1,1)  (3,3,2,1)  (3,2,2,1)  (3,2,1,1)
%t A337505 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A337505 Table[Length[Select[Join@@Permutations/@allnorm[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,3}]
%o A337505 (PARI) \\ here b(n) is A005649.
%o A337505 b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
%o A337505 a(n) = {b(n)*binomial(2*n-1,n)} \\ _Andrew Howroyd_, Dec 31 2020
%Y A337505 A336108 is the version for compositions and runs.
%Y A337505 A337504 is the version for compositions.
%Y A337505 A337506 has this as main diagonal n = 2*k.
%Y A337505 A337564 is the version for runs.
%Y A337505 A000670 counts sequences covering an initial interval.
%Y A337505 A003242 counts anti-run compositions.
%Y A337505 A005649 counts anti-runs covering an initial interval.
%Y A337505 A124767 counts maximal runs in standard compositions.
%Y A337505 A333381 counts maximal anti-runs in standard compositions.
%Y A337505 A333769 gives run-lengths in standard compositions.
%Y A337505 A337565 gives anti-run lengths in standard compositions.
%Y A337505 Cf. A052841, A106351, A106356, A269134, A325535, A333489, A333627, A333755.
%K A337505 nonn
%O A337505 0,2
%A A337505 _Gus Wiseman_, Sep 05 2020
%E A337505 Terms a(5) and beyond from _Andrew Howroyd_, Dec 31 2020