A329744 - OEIS

%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