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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.
58
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
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
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.
Sequence in context: A128065 A364933 A361158 * A277889 A018194 A338630
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Nov 21 2019
STATUS
approved