A329863 Number of compositions of n with cuts-resistance 2.

0, 0, 1, 0, 3, 6, 9, 22, 47, 88, 179, 354, 691, 1344, 2617, 5042, 9709, 18632, 35639, 68010, 129556

Offset

0,5

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 to be the number of applications required to reach an empty word.

Links

Table of n, a(n) for n=0..20.

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

Example

The a(2) = 1 through a(7) = 22 compositions (empty column not shown):
  (1,1)  (2,2)    (1,1,3)    (3,3)      (1,1,5)
         (1,1,2)  (1,2,2)    (1,1,4)    (1,3,3)
         (2,1,1)  (2,2,1)    (4,1,1)    (2,2,3)
                  (3,1,1)    (1,1,2,2)  (3,2,2)
                  (1,1,2,1)  (1,1,3,1)  (3,3,1)
                  (1,2,1,1)  (1,2,2,1)  (5,1,1)
                             (1,3,1,1)  (1,1,2,3)
                             (2,1,1,2)  (1,1,3,2)
                             (2,2,1,1)  (1,1,4,1)
                                        (1,4,1,1)
                                        (2,1,1,3)
                                        (2,1,2,2)
                                        (2,2,1,2)
                                        (2,3,1,1)
                                        (3,1,1,2)
                                        (3,2,1,1)
                                        (1,1,2,1,2)
                                        (1,1,2,2,1)
                                        (1,2,1,1,2)
                                        (1,2,2,1,1)
                                        (2,1,1,2,1)
                                        (2,1,2,1,1)

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], degdep[#]==2&]], {n, 0, 10}]

Crossrefs

Column k = 2 of A329861.

Compositions with cuts-resistance 1 are A003242.

Compositions with runs-resistance 2 are A329745.

Numbers whose binary expansion has cuts-resistance 2 are A329862.

Binary words with cuts-resistance 2 are conjectured to be A027383.

Cuts-resistance of binary expansion is A319416.

Binary words counted by cuts-resistance are A319421 and A329860.

Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.

Sequence in context: A215666 A050889 A327140 * A026095 A061929 A358403

Adjacent sequences: A329860 A329861 A329862 * A329864 A329865 A329866

Keyword

nonn,more

Author

Gus Wiseman, Nov 23 2019

Status

approved