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%I #5 Nov 21 2019 10:43:41
%S 1,1,1,1,1,2,1,2,3,2,1,1,6,6,2,1,3,15,9,4,0,1,1,22,22,16,2,0,1,3,41,
%T 38,37,8,0,0,1,2,72,69,86,26,0,0,0,1,3,129,124,175,78,2,0,0,0,1,1,213,
%U 226,367,202,14,0,0,0,0,1,5,395,376,750,469,52,0,0,0,0,0
%N Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.
%C A composition of n is a finite sequence of positive integers with sum n.
%C For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.
%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e Triangle begins:
%e 1
%e 1 1
%e 1 1 2
%e 1 2 3 2
%e 1 1 6 6 2
%e 1 3 15 9 4 0
%e 1 1 22 22 16 2 0
%e 1 3 41 38 37 8 0 0
%e 1 2 72 69 86 26 0 0 0
%e 1 3 129 124 175 78 2 0 0 0
%e 1 1 213 226 367 202 14 0 0 0 0
%e 1 5 395 376 750 469 52 0 0 0 0 0
%e Row n = 6 counts the following compositions:
%e (6) (33) (15) (114) (1131)
%e (222) (24) (411) (1311)
%e (111111) (42) (1113) (11121)
%e (51) (1221) (12111)
%e (123) (2112)
%e (132) (3111)
%e (141) (11112)
%e (213) (11211)
%e (231) (21111)
%e (312)
%e (321)
%e (1122)
%e (1212)
%e (2121)
%e (2211)
%t runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]
%Y Row sums are A000079.
%Y Column k = 1 is A032741.
%Y Column k = 2 is A329745.
%Y Column k = n - 2 is A329743.
%Y The version for partitions is A329746.
%Y The version with rows reversed is A329750.
%Y Cf. A000740, A008965, A098504, A242882, A318928, A329747.
%K nonn,tabl
%O 1,6
%A _Gus Wiseman_, Nov 21 2019
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