A329861 Triangle read by rows where T(n,k) is the number of compositions of n with cuts-resistance k.

1, 0, 1, 0, 1, 1, 0, 3, 0, 1, 0, 4, 3, 0, 1, 0, 7, 6, 2, 0, 1, 0, 14, 9, 6, 2, 0, 1, 0, 23, 22, 10, 6, 2, 0, 1, 0, 39, 47, 22, 10, 7, 2, 0, 1, 0, 71, 88, 52, 24, 10, 8, 2, 0, 1, 0, 124, 179, 101, 59, 26, 11, 9, 2, 0, 1, 0, 214, 354, 220, 112, 71, 28, 12, 10, 2, 0, 1

Offset

0,8

Comments

A composition of n is a finite sequence of positive integers summing to n.

For the operation of shortening all runs by 1, cuts-resistance is defined as the number of applications required to reach an empty word.

Links

Table of n, a(n) for n=0..77.

Example

Triangle begins:
  1
  0  1
  0  1  1
  0  3  0  1
  0  4  3  0  1
  0  7  6  2  0  1
  0 14  9  6  2  0  1
  0 23 22 10  6  2  0  1
  0 39 47 22 10  7  2  0  1
  0 71 88 52 24 10  8  2  0  1
Row n = 6 counts the following compositions (empty columns not shown):
  (6)     (33)    (222)    (11112)  (111111)
  (15)    (114)   (1113)   (21111)
  (24)    (411)   (3111)
  (42)    (1122)  (11121)
  (51)    (1131)  (11211)
  (123)   (1221)  (12111)
  (132)   (1311)
  (141)   (2112)
  (213)   (2211)
  (231)
  (312)
  (321)
  (1212)
  (2121)

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], degdep[#]==k&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

Row sums are A000079.

Column k = 1 is A003242 (for n > 0).

Column k = 2 is A329863.

Row sums without the k = 1 column are A261983.

The version for runs-resistance is A329744.

The version for binary vectors is A329860.

The cuts-resistance of the binary expansion of n is A319416.

Cf. A175342, A240085, A319411, A319420, A319421, A329738, A329862, A329865.

Sequence in context: A100257 A318315 A373951 * A331332 A300228 A100573

Adjacent sequences: A329858 A329859 A329860 * A329862 A329863 A329864

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 23 2019

Status

approved