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

Author

Gus Wiseman, Nov 23 2019

Keywords

• nonn
• tabl

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.

Examples

			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)
		

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.

Programs

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}]