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A337564 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal runs.

%I A337564 #12 Dec 31 2020 17:01:01
%S A337564 1,1,6,80,1540,38808,1206744,44595408,1908389340,92780281880,
%T A337564 5050066185736,304196411024688,20087958167374552,1442953024024996400,
%U A337564 112007566256683719600,9342904053303870936480,833388624898522799682780,79159669418651567937733080
%N A337564 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal runs.
%C A337564 Sequences covering an initial interval of positive integers are counted by A000670 and ranked by A333217.
%H A337564 Andrew Howroyd, <a href="/A337564/b337564.txt">Table of n, a(n) for n = 0..200</a>
%F A337564 a(n) = A005649(n-1)*binomial(2*n-1,n-1) = A005649(n-1)*A001700(n-1) for n > 0. - _Andrew Howroyd_, Dec 31 2020
%e A337564 The a(0) = 1 through a(2) = 6 sequences:
%e A337564   ()  (1,1)  (1,1,1,2)
%e A337564              (1,1,2,2)
%e A337564              (1,2,2,2)
%e A337564              (2,1,1,1)
%e A337564              (2,2,1,1)
%e A337564              (2,2,2,1)
%e A337564 The a(3) = 80 sequences:
%e A337564   212222  111121  122233  333112  211133
%e A337564   221222  111211  133222  333211  233111
%e A337564   222122  112111  222133  112233  331112
%e A337564   222212  121111  222331  113322  332111
%e A337564   122221  123333  331222  221133  111223
%e A337564   211222  133332  332221  223311  111322
%e A337564   221122  213333  122223  331122  221113
%e A337564   222112  233331  132222  332211  223111
%e A337564   112221  333312  222213  112223  311122
%e A337564   122211  333321  222231  113222  322111
%e A337564   211122  122333  312222  222113  111123
%e A337564   221112  133322  322221  222311  111132
%e A337564   111221  221333  112333  311222  211113
%e A337564   112211  223331  113332  322211  231111
%e A337564   122111  333122  211333  111233  311112
%e A337564   211112  333221  233311  111332  321111
%t A337564 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A337564 Table[Length[Select[Join@@Permutations/@allnorm[2*n],Length[Split[#]]==n&]],{n,0,3}]
%o A337564 (PARI) \\ here b(n) is A005649.
%o A337564 b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
%o A337564 a(n) = {if(n==0, 1, b(n-1)*binomial(2*n-1,n-1))} \\ _Andrew Howroyd_, Dec 31 2020
%Y A337564 A335461 has this as main diagonal n = 2*k.
%Y A337564 A336108 is the version for compositions.
%Y A337564 A337504 is the version for compositions and anti-runs.
%Y A337564 A337505 is the version for anti-runs.
%Y A337564 A000670 counts sequences covering an initial interval.
%Y A337564 A005649 counts anti-runs covering an initial interval.
%Y A337564 A124767 counts maximal runs in standard compositions.
%Y A337564 A333769 gives run lengths in standard compositions.
%Y A337564 A337504 counts compositions of 2*n with n maximal anti-runs.
%Y A337564 A337565 gives anti-run lengths in standard compositions.
%Y A337564 Cf. A001700, A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A333755, A335838.
%K A337564 nonn
%O A337564 0,3
%A A337564 _Gus Wiseman_, Sep 03 2020
%E A337564 Terms a(5) and beyond from _Andrew Howroyd_, Dec 31 2020