A330028 Number of compositions of n with cuts-resistance <= 2.

1, 1, 2, 3, 7, 13, 23, 45, 86, 159, 303, 568, 1069, 2005, 3769, 7066, 13251, 24821, 46482, 86988, 162758

Offset: 0

Author

Gus Wiseman, Nov 27 2019

Keywords

• nonn
• more

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.

Examples

			The a(0) = 1 through a(5) = 13 compositions:
  ()  (1)  (2)    (3)    (4)      (5)
           (1,1)  (1,2)  (1,3)    (1,4)
                  (2,1)  (2,2)    (2,3)
                         (3,1)    (3,2)
                         (1,1,2)  (4,1)
                         (1,2,1)  (1,1,3)
                         (2,1,1)  (1,2,2)
                                  (1,3,1)
                                  (2,1,2)
                                  (2,2,1)
                                  (3,1,1)
                                  (1,1,2,1)
                                  (1,2,1,1)
		

Crossrefs

Sum of first three columns of A329861.

Compositions with cuts-resistance 1 are A003242.

Compositions with cuts-resistance 2 are A329863.

Compositions with runs-resistance 2 are A329745.

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

Binary words with cuts-resistance 2 are A027383.

Cuts-resistance of binary expansion is A319416.

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

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

Programs

Mathematica

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