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A329869
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Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.
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3
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0, 1, 2, 1, 2, 1, 4, 5, 11, 19, 36, 77, 138, 252, 528, 1072, 2204, 4634, 9575, 19732, 40754
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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EXAMPLE
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The a(1) = 1 through a(9) = 19 compositions:
1 2 3 4 5 6 7 8 9
11 22 33 11113 44 11115
11112 31111 11114 12222
21111 111211 41111 22221
112111 111122 51111
111311 111222
113111 111411
211112 114111
221111 211113
1111121 222111
1211111 311112
1111131
1111221
1112112
1121112
1221111
1311111
2111211
2112111
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).
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MATHEMATICA
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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[#]+1==degdep[#]&]], {n, 0, 10}]
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CROSSREFS
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The version for binary indices is A329866.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Cf. A003242, A098504, A114901, A242882, A318928, A319411, A319416, A319420, A319421, A329864, A329865, A329867, A329868.
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KEYWORD
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nonn,more
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AUTHOR
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STATUS
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approved
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