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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. 55
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, 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, 226, 367, 202, 14, 0, 0, 0, 0, 1, 5, 395, 376, 750, 469, 52, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

A composition of n is a finite sequence of positive integers with sum n.

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.

LINKS

Table of n, a(n) for n=1..78.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

EXAMPLE

Triangle begins:

   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  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   1 213 226 367 202  14   0   0   0   0

   1   5 395 376 750 469  52   0   0   0   0   0

Row n = 6 counts the following compositions:

  (6)  (33)      (15)    (114)    (1131)

       (222)     (24)    (411)    (1311)

       (111111)  (42)    (1113)   (11121)

                 (51)    (1221)   (12111)

                 (123)   (2112)

                 (132)   (3111)

                 (141)   (11112)

                 (213)   (11211)

                 (231)   (21111)

                 (312)

                 (321)

                 (1122)

                 (1212)

                 (2121)

                 (2211)

MATHEMATICA

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==k&]], {n, 10}, {k, 0, n-1}]

CROSSREFS

Row sums are A000079.

Column k = 1 is A032741.

Column k = 2 is A329745.

Column k = n - 2 is A329743.

The version for partitions is A329746.

The version with rows reversed is A329750.

Cf. A000740, A008965, A098504, A242882, A318928, A329747.

Sequence in context: A243919 A231775 A128065 * A277889 A018194 A338630

Adjacent sequences:  A329741 A329742 A329743 * A329745 A329746 A329747

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Nov 21 2019

STATUS

approved

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Last modified May 11 16:45 EDT 2021. Contains 343804 sequences. (Running on oeis4.)