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proposed
approved
editing
proposed
Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
allocated for Gus WisemanNumbers whose binary expansion has the same runs-resistance as cuts-resistance.
0, 8, 12, 14, 17, 24, 27, 28, 35, 36, 39, 47, 49, 51, 54, 57, 61, 70, 73, 78, 80, 99, 122, 130, 156, 175, 184, 189, 190, 198, 204, 207, 208, 215, 216, 226, 228, 235, 243, 244, 245, 261, 271, 283, 295, 304, 313, 321, 322, 336, 352, 367, 375, 378, 379, 380, 386
1,2
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.
The sequence of terms together with their binary expansions begins:
0:
8: 1000
12: 1100
14: 1110
17: 10001
24: 11000
27: 11011
28: 11100
35: 100011
36: 100100
39: 100111
47: 101111
49: 110001
51: 110011
54: 110110
57: 111001
61: 111101
70: 1000110
73: 1001001
78: 1001110
80: 1010000
For example, 36 has runs-resistance 3 because we have (100100) -> (1212) -> (1111) -> (4), while the cuts-resistance is also 3 because we have (100100) -> (00) -> (0) -> ().
Similarly, 57 has runs-resistance 3 because we have (111001) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have (111001) -> (110) -> (1) -> ().
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Select[Range[0, 100], #==0||runsres[IntegerDigits[#, 2]]==degdep[IntegerDigits[#, 2]]&]
Positions of 0's in A329867.
Compositions with runs-resistance equal to cuts-resistance are A329864.
The version for runs-resistance equal to cuts-resistance minus 1 is A329866.
The runs-resistance of the binary expansion is A318928.
The cuts-resistance of the binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
Cf. A000975, A003242, A107907, A164707, A319420, A329738, A329868.
allocated
nonn
Gus Wiseman, Nov 23 2019
approved
editing
allocated for Gus Wiseman
allocated
approved