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A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.

%I A337506 #13 Dec 31 2020 19:24:13
%S A337506 1,0,1,0,2,1,0,8,4,1,0,44,24,6,1,0,308,176,48,8,1,0,2612,1540,440,80,
%T A337506 10,1,0,25988,15672,4620,880,120,12,1,0,296564,181916,54852,10780,
%U A337506 1540,168,14,1,0,3816548,2372512,727664,146272,21560,2464,224,16,1
%N A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.
%C A337506 An anti-run is a sequence with no adjacent equal parts. The number of maximal anti-runs is one more than the number of adjacent equal parts.
%H A337506 Andrew Howroyd, <a href="/A337506/b337506.txt">Table of n, a(n) for n = 0..1325</a>
%F A337506 T(n,k) = A005649(n-k) * binomial(n-1,k-1) for k > 0. - _Andrew Howroyd_, Dec 31 2020
%e A337506 Triangle begins:
%e A337506   1
%e A337506   0      1
%e A337506   0      2      1
%e A337506   0      8      4      1
%e A337506   0     44     24      6      1
%e A337506   0    308    176     48      8      1
%e A337506   0   2612   1540    440     80     10      1
%e A337506   0  25988  15672   4620    880    120     12      1
%e A337506   0 296564 181916  54852  10780   1540    168     14      1
%e A337506 Row n = 3 counts the following sequences (empty column indicated by dot):
%e A337506   .  (1,2,1)  (1,1,2)  (1,1,1)
%e A337506      (1,2,3)  (1,2,2)
%e A337506      (1,3,2)  (2,1,1)
%e A337506      (2,1,2)  (2,2,1)
%e A337506      (2,1,3)
%e A337506      (2,3,1)
%e A337506      (3,1,2)
%e A337506      (3,2,1)
%t A337506 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A337506 Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==k&]],{n,0,5},{k,0,n}]
%o A337506 (PARI) \\ here b(n) is A005649.
%o A337506 b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
%o A337506 T(n,k)=if(n==0, k==0, b(n-k)*binomial(n-1,k-1)) \\ _Andrew Howroyd_, Dec 31 2020
%Y A337506 A000670 gives row sums.
%Y A337506 A005649 gives column k = 1.
%Y A337506 A337507 gives column k = 2.
%Y A337506 A337505 gives the diagonal n = 2*k.
%Y A337506 A106356 is the version for compositions.
%Y A337506 A238130/A238279/A333755 is the version for runs in compositions.
%Y A337506 A335461 has the reversed rows (except zeros).
%Y A337506 A003242 counts anti-run compositions.
%Y A337506 A124762 counts adjacent equal terms in standard compositions.
%Y A337506 A124767 counts maximal runs in standard compositions.
%Y A337506 A333381 counts maximal anti-runs in standard compositions.
%Y A337506 A333382 counts adjacent unequal terms in standard compositions.
%Y A337506 A333489 ranks anti-run compositions.
%Y A337506 A333769 gives maximal run-lengths in standard compositions.
%Y A337506 A337565 gives maximal anti-run lengths in standard compositions.
%Y A337506 Cf. A019472, A052841, A060223, A106351, A269134, A325535, A337564.
%K A337506 nonn,tabl
%O A337506 0,5
%A A337506 _Gus Wiseman_, Sep 06 2020
%E A337506 Terms a(45) and beyond from _Andrew Howroyd_, Dec 31 2020