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%I A333755 #12 Apr 10 2020 08:16:10
%S A333755 1,0,1,0,2,0,0,2,2,0,0,3,4,1,0,0,2,10,4,0,0,0,4,12,14,2,0,0,0,2,22,29,
%T A333755 10,1,0,0,0,4,26,56,36,6,0,0,0,0,3,34,100,86,31,2,0,0,0,0,4,44,148,
%U A333755 200,99,16,1,0,0,0,0,2,54,230,374,278,78,8,0,0,0,0
%N A333755 Triangle read by rows where T(n,k) is the number of compositions of n with k runs, n >= 0, 0 <= k <= n.
%C A333755 Except for a(1) = 0, the data is identical to A238130 shifted right once. However, in A238130, each row after the first ends with a zero, while here each row after the first starts with a zero.
%e A333755 Triangle begins:
%e A333755    1
%e A333755    0   1
%e A333755    0   2   0
%e A333755    0   2   2   0
%e A333755    0   3   4   1   0
%e A333755    0   2  10   4   0   0
%e A333755    0   4  12  14   2   0   0
%e A333755    0   2  22  29  10   1   0   0
%e A333755    0   4  26  56  36   6   0   0   0
%e A333755    0   3  34 100  86  31   2   0   0   0
%e A333755    0   4  44 148 200  99  16   1   0   0   0
%e A333755    0   2  54 230 374 278  78   8   0   0   0   0
%e A333755 Row n = 6 counts the following compositions (empty column indicated by dot):
%e A333755   .  (6)       (15)     (123)    (1212)
%e A333755      (33)      (24)     (132)    (2121)
%e A333755      (222)     (42)     (141)
%e A333755      (111111)  (51)     (213)
%e A333755                (114)    (231)
%e A333755                (411)    (312)
%e A333755                (1113)   (321)
%e A333755                (1122)   (1131)
%e A333755                (2211)   (1221)
%e A333755                (3111)   (1311)
%e A333755                (11112)  (2112)
%e A333755                (21111)  (11121)
%e A333755                         (11211)
%e A333755                         (12111)
%t A333755 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Length[Split[#]]==k&]],{n,0,10},{k,0,n}]
%Y A333755 Removing all zeros gives A238279.
%Y A333755 The version for anti-runs is A106356.
%Y A333755 The k-th composition in standard-order has A124767(k) runs.
%Y A333755 The version counting descents is A238343.
%Y A333755 The version counting weak ascents is A333213.
%Y A333755 Cf. A066099, A124762, A238130, A272919, A333381, A333382, A333489.
%K A333755 nonn,tabl
%O A333755 0,5
%A A333755 _Gus Wiseman_, Apr 10 2020

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