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%I A329869 #4 Nov 24 2019 10:00:52
%S A329869 0,1,2,1,2,1,4,5,11,19,36,77,138,252,528,1072,2204,4634,9575,19732,
%T A329869 40754
%N A329869 Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.
%C A329869 A composition of n is a finite sequence of positive integers summing to n.
%C A329869 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 A329869 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 A329869 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329869 The a(1) = 1 through a(9) = 19 compositions:
%e A329869   1   2   3   4   5   6      7       8        9
%e A329869       11      22      33     11113   44       11115
%e A329869                       11112  31111   11114    12222
%e A329869                       21111  111211  41111    22221
%e A329869                              112111  111122   51111
%e A329869                                      111311   111222
%e A329869                                      113111   111411
%e A329869                                      211112   114111
%e A329869                                      221111   211113
%e A329869                                      1111121  222111
%e A329869                                      1211111  311112
%e A329869                                               1111131
%e A329869                                               1111221
%e A329869                                               1112112
%e A329869                                               1121112
%e A329869                                               1221111
%e A329869                                               1311111
%e A329869                                               2111211
%e A329869                                               2112111
%e A329869 For example, the runs-resistance of (1221111) is 3 because we have: (1221111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have: (1221111) -> (2111) -> (11) -> (1) -> (), so (1221111) is counted under a(9).
%t A329869 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329869 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329869 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]+1==degdep[#]&]],{n,0,10}]
%Y A329869 The version for binary indices is A329866.
%Y A329869 Compositions counted by runs-resistance are A329744.
%Y A329869 Compositions counted by cuts-resistance are A329861.
%Y A329869 Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329864, A329865, A329867, A329868.
%K A329869 nonn,more
%O A329869 0,3
%A A329869 _Gus Wiseman_, Nov 23 2019

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