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 A329864 Number of compositions of n with the same runs-resistance as cuts-resistance. 7
 1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS A composition of n is a finite sequence of positive integers summing to n. 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. 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(5) = 2 through a(8) = 17 compositions:   (1112)  (1113)   (1114)    (1115)   (2111)  (1122)   (1222)    (1133)           (2211)   (2221)    (3311)           (3111)   (4111)    (5111)           (11211)  (11122)   (11222)                    (11311)   (11411)                    (21112)   (12221)                    (22111)   (21113)                    (111121)  (22211)                    (121111)  (31112)                              (111131)                              (111221)                              (112112)                              (112211)                              (122111)                              (131111)                              (211211) 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). MATHEMATICA runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1; degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1; Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==degdep[#]&]], {n, 0, 10}] CROSSREFS The version for binary expansion is A329865. Compositions counted by runs-resistance are A329744. Compositions counted by cuts-resistance are A329861. Compositions with runs-resistance = cuts-resistance minus 1 are A329869. Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329867, A329868. Sequence in context: A062493 A056871 A246883 * A174910 A301273 A007504 Adjacent sequences:  A329861 A329862 A329863 * A329865 A329866 A329867 KEYWORD nonn,more AUTHOR Gus Wiseman, Nov 23 2019 STATUS approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)