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Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.
+20
5
0, 1, 2, 7, 11, 15, 18, 31, 63, 75, 127, 255, 511, 1023, 1234, 2047, 4095, 8191, 9638, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607
COMMENTS
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.
EXAMPLE
The sequence of terms together with their binary expansions begins:
0:
1: 1
2: 10
7: 111
11: 1011
15: 1111
18: 10010
31: 11111
63: 111111
75: 1001011
127: 1111111
255: 11111111
511: 111111111
1023: 1111111111
1234: 10011010010
2047: 11111111111
4095: 111111111111
8191: 1111111111111
9638: 10010110100110
16383: 11111111111111
32767: 111111111111111
65535: 1111111111111111
MATHEMATICA
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
das=Table[If[n==0, 0, runsres[IntegerDigits[n, 2]]-degdep[IntegerDigits[n, 2]]], {n, 0, 1000000}];
Table[Position[das, i][[1, 1]]-1, {i, First/@Gather[das]}]
CROSSREFS
Sorted positions of first appearances in A329867. Compositions with runs-resistance equal to cuts-resistance are A329864. Runs-resistance of binary expansion is A318928. Cuts-resistance of binary expansion is A319416. Compositions counted by runs-resistance are A329744. Compositions counted by cuts-resistance are A329861.
Runs-resistance of binary representation of n.
+10
65
1, 2, 1, 3, 2, 3, 1, 3, 3, 2, 4, 2, 4, 3, 1, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 4, 3, 4, 5, 2, 4, 3, 4, 5, 4, 3, 3, 3, 2, 4, 4, 3, 3, 2, 3, 4, 3, 3, 4, 3, 1, 3, 3, 5, 3, 3, 5, 3, 4, 3, 3, 5, 6, 4, 5, 3, 3, 4, 5, 4, 4, 4, 2, 5, 4, 5, 5, 4, 5, 5, 4, 5, 4
COMMENTS
Following Lenormand (2003), we define the "runs-resistance" of a finite list L to be the number of times the RUNS transformation must be applied to L in order to reduce L to a list with a single element.
Here it is immaterial whether we read the binary representation of n from left to right or right to left.
The RUNS transformation must be applied at least once, in order to obtain a list, so a(n) >= 1.
LINKS
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003.
EXAMPLE
11 in binary is [1, 0, 1, 1],
which has runs of lengths [1, 1, 2],
which has runs of lengths [2, 1],
which has runs of lengths [1, 1],
which has a single run of length [2].
This took four steps, so a(11) = 4.
MAPLE
with(transforms);
# compute Lenormand's "resistance" of a list
resist:=proc(a) local ct, i, b;
if whattype(a) <> list then ERROR("input must be a list"); fi:
ct:=0; b:=a; for i from 1 to 100 do
if nops(b)=1 then return(ct); fi;
b:=RUNS(b); ct:=ct+1; od; end;
a:=[1];
for n from 2 to 100 do
b:=convert(n, base, 2);
r:=resist(b);
a:=[op(a), r];
od:
MATHEMATICA
Table[If[n == 1, 1, Length[NestWhileList[Length/@Split[#] &, IntegerDigits[n, 2], Length[#] > 1 &]] - 1], {n, 50}] (* Gus Wiseman, Nov 25 2019 *)
CROSSREFS
Ignoring the first digit gives A329870. Compositions counted by runs-resistance are A329744.
Numbers whose binary expansion has the same runs-resistance as cuts-resistance.
+10
14
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
COMMENTS
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.
EXAMPLE
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) -> ().
MATHEMATICA
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]]&]
CROSSREFS
The version for runs-resistance equal to cuts-resistance minus 1 is A329866. Compositions with runs-resistance equal to cuts-resistance are A329864. Runs-resistance of binary expansion is A318928. Cuts-resistance of binary expansion is A319416. Compositions counted by runs-resistance are A329744. Compositions counted by cuts-resistance are A329861.
Number of compositions of n with the same runs-resistance as cuts-resistance.
+10
7
1, 0, 0, 0, 0, 2, 5, 10, 17, 27, 68, 107, 217, 420, 884, 1761, 3679, 7469, 15437, 31396, 64369
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.
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.
Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.
+10
5
1, 3, 16, 30, 33, 48, 55, 56, 59, 60, 67, 68, 72, 79, 95, 97, 110, 112, 118, 120, 121, 125, 134, 135, 137, 143, 145, 158, 160, 195, 196, 219, 220, 225, 231, 241, 250, 258, 270, 280, 286, 291, 292, 315, 316, 351, 381, 382, 390, 391, 393, 399, 415, 416, 431, 432
COMMENTS
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.
EXAMPLE
The sequence of terms together with their binary expansions begins:
1: 1
3: 11
16: 10000
30: 11110
33: 100001
48: 110000
55: 110111
56: 111000
59: 111011
60: 111100
67: 1000011
68: 1000100
72: 1001000
79: 1001111
95: 1011111
97: 1100001
110: 1101110
112: 1110000
118: 1110110
120: 1111000
For example, 79 has runs-resistance 3 because we have (1001111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have (1001111) -> (0111) -> (11) -> (1) -> (), so 79 is in the sequence.
MATHEMATICA
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Select[Range[100], runsres[IntegerDigits[#, 2]]-degdep[IntegerDigits[#, 2]]==-1&]
CROSSREFS
The version for runs-resistance equal to cuts-resistance is A329865. Compositions with runs-resistance equal to cuts-resistance are A329864. Compositions with runs-resistance = cuts-resistance minus 1 are A329869. Runs-resistance of binary expansion is A318928. Cuts-resistance of binary expansion is A319416. Compositions counted by runs-resistance are A329744. Compositions counted by cuts-resistance are A329861.
Number of compositions of n with runs-resistance equal to cuts-resistance minus 1.
+10
3
0, 1, 2, 1, 2, 1, 4, 5, 11, 19, 36, 77, 138, 252, 528, 1072, 2204, 4634, 9575, 19732, 40754
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.
EXAMPLE
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).
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[#]+1==degdep[#]&]], {n, 0, 10}]
CROSSREFS
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.
Runs-resistance of the binary expansion of n without the first digit.
+10
3
0, 0, 1, 2, 2, 1, 1, 3, 2, 3, 3, 2, 3, 1, 1, 3, 4, 2, 4, 2, 3, 3, 3, 3, 2, 4, 2, 4, 3, 1, 1, 3, 4, 3, 3, 4, 4, 3, 4, 5, 2, 4, 4, 5, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 1, 3, 4, 3, 3, 4, 3, 2, 3, 3, 4, 4, 2, 3, 3, 3, 4, 5, 4, 3, 4, 2, 5, 4
COMMENTS
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.
EXAMPLE
Minimal representatives with each image are:
2: (0)
4: (0,0) -> (2)
5: (0,1) -> (1,1) -> (2)
9: (0,0,1) -> (2,1) -> (1,1) -> (2)
18: (0,0,1,0) -> (2,1,1) -> (1,2) -> (1,1) -> (2)
41: (0,1,0,0,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2)
150: (0,0,1,0,1,1,0) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1) -> (2)
MATHEMATICA
Table[Length[NestWhileList[Length/@Split[#]&, Rest[IntegerDigits[n, 2]], Length[#]>1&]]-1, {n, 2, 100}]
CROSSREFS
Keeping the first digit gives A318928. Compositions counted by runs-resistance are A329744.
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