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
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.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 23 2019
STATUS
approved