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A329744 Triangle read by rows where T(n,k) is the number of compositions of n > 0 with runs-resistance k, 0 <= k <= n - 1.

%I A329744 #5 Nov 21 2019 10:43:41
%S A329744 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 A329744 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 A329744 226,367,202,14,0,0,0,0,1,5,395,376,750,469,52,0,0,0,0,0
%N A329744 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 A329744 A composition of n is a finite sequence of positive integers with sum n.
%C A329744 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 A329744 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329744 Triangle begins:
%e A329744    1
%e A329744    1   1
%e A329744    1   1   2
%e A329744    1   2   3   2
%e A329744    1   1   6   6   2
%e A329744    1   3  15   9   4   0
%e A329744    1   1  22  22  16   2   0
%e A329744    1   3  41  38  37   8   0   0
%e A329744    1   2  72  69  86  26   0   0   0
%e A329744    1   3 129 124 175  78   2   0   0   0
%e A329744    1   1 213 226 367 202  14   0   0   0   0
%e A329744    1   5 395 376 750 469  52   0   0   0   0   0
%e A329744 Row n = 6 counts the following compositions:
%e A329744   (6)  (33)      (15)    (114)    (1131)
%e A329744        (222)     (24)    (411)    (1311)
%e A329744        (111111)  (42)    (1113)   (11121)
%e A329744                  (51)    (1221)   (12111)
%e A329744                  (123)   (2112)
%e A329744                  (132)   (3111)
%e A329744                  (141)   (11112)
%e A329744                  (213)   (11211)
%e A329744                  (231)   (21111)
%e A329744                  (312)
%e A329744                  (321)
%e A329744                  (1122)
%e A329744                  (1212)
%e A329744                  (2121)
%e A329744                  (2211)
%t A329744 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329744 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==k&]],{n,10},{k,0,n-1}]
%Y A329744 Row sums are A000079.
%Y A329744 Column k = 1 is A032741.
%Y A329744 Column k = 2 is A329745.
%Y A329744 Column k = n - 2 is A329743.
%Y A329744 The version for partitions is A329746.
%Y A329744 The version with rows reversed is A329750.
%Y A329744 Cf. A000740, A008965, A098504, A242882, A318928, A329747.
%K A329744 nonn,tabl
%O A329744 1,6
%A A329744 _Gus Wiseman_, Nov 21 2019