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A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.

%I A335461 #9 Dec 31 2020 17:02:00
%S A335461 1,0,1,0,1,2,0,1,4,8,0,1,6,24,44,0,1,8,48,176,308,0,1,10,80,440,1540,
%T A335461 2612,0,1,12,120,880,4620,15672,25988,0,1,14,168,1540,10780,54852,
%U A335461 181916,296564,0,1,16,224,2464,21560,146272,727664,2372512,3816548
%N A335461 Triangle read by rows where T(n,k) is the number of patterns of length n with k runs.
%C A335461 We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
%H A335461 Andrew Howroyd, <a href="/A335461/b335461.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A335461 T(n,k) = A005649(k-1) * binomial(n-1,k-1) for k > 0. - _Andrew Howroyd_, Dec 31 2020
%e A335461 Triangle begins:
%e A335461      1
%e A335461      0     1
%e A335461      0     1     2
%e A335461      0     1     4     8
%e A335461      0     1     6    24    44
%e A335461      0     1     8    48   176   308
%e A335461      0     1    10    80   440  1540  2612
%e A335461      0     1    12   120   880  4620 15672 25988
%e A335461 Row n = 3 counts the following patterns:
%e A335461   (1,1,1)  (1,1,2)  (1,2,1)
%e A335461            (1,2,2)  (1,2,3)
%e A335461            (2,1,1)  (1,3,2)
%e A335461            (2,2,1)  (2,1,2)
%e A335461                     (2,1,3)
%e A335461                     (2,3,1)
%e A335461                     (3,1,2)
%e A335461                     (3,2,1)
%t A335461 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A335461 Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#]]==k&]],{n,0,5},{k,0,n}]
%o A335461 (PARI) \\ here b(n) is A005649.
%o A335461 b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
%o A335461 T(n,k)=if(n==0, k==0, b(k-1)*binomial(n-1,k-1)) \\ _Andrew Howroyd_, Dec 31 2020
%Y A335461 Row sums are A000670.
%Y A335461 Column n = k is A005649 (anti-run patterns).
%Y A335461 Central coefficients are A337564.
%Y A335461 The version for compositions is A333755.
%Y A335461 Runs of standard compositions are counted by A124767.
%Y A335461 Run-lengths of standard compositions are A333769.
%Y A335461 Cf. A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A335838.
%K A335461 nonn,tabl
%O A335461 0,6
%A A335461 _Gus Wiseman_, Jul 03 2020