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Number of compositions of n with cuts-resistance <= 2.
1

%I #8 Nov 28 2019 08:07:10

%S 1,1,2,3,7,13,23,45,86,159,303,568,1069,2005,3769,7066,13251,24821,

%T 46482,86988,162758

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

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

%C 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.

%e The a(0) = 1 through a(5) = 13 compositions:

%e () (1) (2) (3) (4) (5)

%e (1,1) (1,2) (1,3) (1,4)

%e (2,1) (2,2) (2,3)

%e (3,1) (3,2)

%e (1,1,2) (4,1)

%e (1,2,1) (1,1,3)

%e (2,1,1) (1,2,2)

%e (1,3,1)

%e (2,1,2)

%e (2,2,1)

%e (3,1,1)

%e (1,1,2,1)

%e (1,2,1,1)

%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;

%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]<=2&]],{n,0,10}]

%Y Sum of first three columns of A329861.

%Y Compositions with cuts-resistance 1 are A003242.

%Y Compositions with cuts-resistance 2 are A329863.

%Y Compositions with runs-resistance 2 are A329745.

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

%Y Binary words with cuts-resistance 2 are A027383.

%Y Cuts-resistance of binary expansion is A319416.

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

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

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Nov 27 2019