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