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A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.

%I A329864 #8 Nov 24 2019 10:00:09
%S A329864 1,0,0,0,0,2,5,10,17,27,68,107,217,420,884,1761,3679,7469,15437,31396,
%T A329864 64369
%N A329864 Number of compositions of n with the same runs-resistance as cuts-resistance.
%C A329864 A composition of n is a finite sequence of positive integers summing to n.
%C A329864 For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
%C A329864 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.
%H A329864 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329864 The a(5) = 2 through a(8) = 17 compositions:
%e A329864   (1112)  (1113)   (1114)    (1115)
%e A329864   (2111)  (1122)   (1222)    (1133)
%e A329864           (2211)   (2221)    (3311)
%e A329864           (3111)   (4111)    (5111)
%e A329864           (11211)  (11122)   (11222)
%e A329864                    (11311)   (11411)
%e A329864                    (21112)   (12221)
%e A329864                    (22111)   (21113)
%e A329864                    (111121)  (22211)
%e A329864                    (121111)  (31112)
%e A329864                              (111131)
%e A329864                              (111221)
%e A329864                              (112112)
%e A329864                              (112211)
%e A329864                              (122111)
%e A329864                              (131111)
%e A329864                              (211211)
%e A329864 For example, the runs-resistance of (111221) is 3 because we have: (111221) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have: (111221) -> (112) -> (1) -> (), so (111221) is counted under a(8).
%t A329864 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329864 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329864 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==degdep[#]&]],{n,0,10}]
%Y A329864 The version for binary expansion is A329865.
%Y A329864 Compositions counted by runs-resistance are A329744.
%Y A329864 Compositions counted by cuts-resistance are A329861.
%Y A329864 Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
%Y A329864 Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868.
%K A329864 nonn,more
%O A329864 0,6
%A A329864 _Gus Wiseman_, Nov 23 2019