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Search: a329869 -id:a329869
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A329864
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
(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
Table of n, a(n) for n=0..20.
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.
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 23 2019
STATUS
approved
A329867
Runs-resistance minus cuts-resistance of the binary expansion of n.
+10
7
0, -1, 1, -1, 1, 1, 1, -2, 0, 1, 1, 2, 0, 2, 0, -3, -1, 0, 3, 2, 2, 1, 3, 1, 0, 2, 2, 0, 0, 1, -1, -4, -2, -1, 2, 0, 0, 3, 2, 0, 1, 3, 1, 2, 1, 2, 2, 0, -1, 0, 1, 0, 2, 2, 0, -1, -1, 0, 1, -1, -1, 0, -2, -5, -3, -2, 1, -1, -1, 2, 0, 1, -1, 0, 3, 4, 2, 3, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,8
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.
LINKS
Table of n, a(n) for n=0..78.
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
FORMULA
For n > 1, a(2^n) = 3 - n.
For n > 1, a(2^n - 1) = 1 - n.
EXAMPLE
The sequence of binary expansions together with their runs-resistances and cuts-resistances, and their differences, begins:
0 (): 0 - 0 = 0
1 (1): 0 - 1 = -1
2 (10): 2 - 1 = 1
3 (11): 1 - 2 = -1
4 (100): 3 - 2 = 1
5 (101): 2 - 1 = 1
6 (110): 3 - 2 = 1
7 (111): 1 - 3 = -2
8 (1000): 3 - 3 = 0
9 (1001): 3 - 2 = 1
10 (1010): 2 - 1 = 1
11 (1011): 4 - 2 = 2
12 (1100): 2 - 2 = 0
13 (1101): 4 - 2 = 2
14 (1110): 3 - 3 = 0
15 (1111): 1 - 4 = -3
16 (10000): 3 - 4 = -1
17 (10001): 3 - 3 = 0
18 (10010): 5 - 2 = 3
19 (10011): 4 - 2 = 2
20 (10100): 4 - 2 = 2
MATHEMATICA
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[If[n==0, 0, runsres[IntegerDigits[n, 2]]-degdep[IntegerDigits[n, 2]]], {n, 0, 100}]
CROSSREFS
Positions of 0's are A329865.
Positions of -1's are A329866.
Sorted positions of first appearances are A329868.
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.
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, A329738.
KEYWORD
sign
AUTHOR
Gus Wiseman, Nov 23 2019
STATUS
approved
A329866
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
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.
LINKS
Table of n, a(n) for n=1..56.
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
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
Positions of -1's in A329867.
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.
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, A329738, A329868.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 23 2019
STATUS
approved
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