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A329746 Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1. 24
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0, 1, 3, 4, 3, 0, 0, 1, 1, 4, 8, 1, 0, 0, 1, 3, 6, 10, 2, 0, 0, 0, 1, 2, 8, 13, 6, 0, 0, 0, 0, 1, 3, 11, 20, 7, 0, 0, 0, 0, 0, 1, 1, 11, 29, 14, 0, 0, 0, 0, 0, 0, 1, 5, 19, 31, 20, 1, 0, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

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  1

  1  2  1  1

  1  1  2  3  0

  1  3  4  3  0  0

  1  1  4  8  1  0  0

  1  3  6 10  2  0  0  0

  1  2  8 13  6  0  0  0  0

  1  3 11 20  7  0  0  0  0  0

  1  1 11 29 14  0  0  0  0  0  0

  1  5 19 31 20  1  0  0  0  0  0  0

  1  1 17 50 30  2  0  0  0  0  0  0  0

  1  3 25 64 37  5  0  0  0  0  0  0  0  0

  1  3 29 74 62  7  0  0  0  0  0  0  0  0  0

Row n = 8 counts the following partitions:

  (8)  (44)        (53)    (332)      (4211)

       (2222)      (62)    (422)      (32111)

       (11111111)  (71)    (611)

                   (431)   (3221)

                   (521)   (5111)

                   (3311)  (22211)

                           (41111)

                           (221111)

                           (311111)

                           (2111111)

MATHEMATICA

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

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

CROSSREFS

Row sums are A000041.

Column k = 1 is A032741.

Column k = 2 is A329745.

A similar invariant is frequency depth; see A323014, A325280.

The version for compositions is A329744.

The version for binary words is A329767.

Cf. A098504, A182850, A225485, A242882, A318928, A325410, A329747.

Sequence in context: A140225 A104758 A143227 * A302247 A026791 A080576

Adjacent sequences:  A329743 A329744 A329745 * A329747 A329748 A329749

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Nov 21 2019

STATUS

approved

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Last modified June 24 14:24 EDT 2021. Contains 345417 sequences. (Running on oeis4.)