Displaying 1-10 of 24 results found.
Numbers k such that every rotation of the binary digits of k is less than k.
+10
62
0, 1, 2, 4, 6, 8, 12, 14, 16, 20, 24, 26, 28, 30, 32, 40, 48, 50, 52, 56, 58, 60, 62, 64, 72, 80, 84, 96, 98, 100, 104, 106, 108, 112, 114, 116, 118, 120, 122, 124, 126, 128, 144, 160, 164, 168, 192, 194, 196, 200, 202, 208, 210, 212, 216, 218, 224, 226, 228
COMMENTS
Number of terms with d binary digits is A001037(d). Take the binary representation of a(n), reverse it, add 1 to each digit. The result is the decimal representation of A102659(n). Also numbers k such that the k-th composition in standard order (row k of A066099) is a Lyndon word. For example, the sequence of all Lyndon words begins: 0: () 52: (1,2,3) 118: (1,1,2,1,2)
1: (1) 56: (1,1,4) 120: (1,1,1,4)
2: (2) 58: (1,1,2,2) 122: (1,1,1,2,2)
4: (3) 60: (1,1,1,3) 124: (1,1,1,1,3)
6: (1,2) 62: (1,1,1,1,2) 126: (1,1,1,1,1,2)
8: (4) 64: (7) 128: (8)
12: (1,3) 72: (3,4) 144: (3,5)
14: (1,1,2) 80: (2,5) 160: (2,6)
16: (5) 84: (2,2,3) 164: (2,3,3)
20: (2,3) 96: (1,6) 168: (2,2,4)
24: (1,4) 98: (1,4,2) 192: (1,7)
26: (1,2,2) 100: (1,3,3) 194: (1,5,2)
28: (1,1,3) 104: (1,2,4) 196: (1,4,3)
30: (1,1,1,2) 106: (1,2,2,2) 200: (1,3,4)
32: (6) 108: (1,2,1,3) 202: (1,3,2,2)
40: (2,4) 112: (1,1,5) 208: (1,2,5)
48: (1,5) 114: (1,1,3,2) 210: (1,2,3,2)
50: (1,3,2) 116: (1,1,2,3) 212: (1,2,2,3)
(End)
EXAMPLE
6 is in the sequence because its binary representation 110 is greater than all the rotations 011 and 101.
10 is not in the sequence because its binary representation 1010 is unchanged under rotation by 2 places.
The sequence of terms together with their binary expansions and binary indices begins:
1: 1 ~ {1}
2: 10 ~ {2}
4: 100 ~ {3}
6: 110 ~ {2,3}
8: 1000 ~ {4}
12: 1100 ~ {3,4}
14: 1110 ~ {2,3,4}
16: 10000 ~ {5}
20: 10100 ~ {3,5}
24: 11000 ~ {4,5}
26: 11010 ~ {2,4,5}
28: 11100 ~ {3,4,5}
30: 11110 ~ {2,3,4,5}
32: 100000 ~ {6}
40: 101000 ~ {4,6}
48: 110000 ~ {5,6}
50: 110010 ~ {2,5,6}
52: 110100 ~ {3,5,6}
56: 111000 ~ {4,5,6}
58: 111010 ~ {2,4,5,6}
(End)
MAPLE
filter:= proc(n) local L, k;
L:= convert(convert(n, binary), string);
for k from 1 to length(L)-1 do
if lexorder(L, StringTools:-Rotate(L, k)) then return false fi;
od;
true
end proc:
select(filter, [$0..1000]);
MATHEMATICA
filterQ[n_] := Module[{bits, rr}, bits = IntegerDigits[n, 2]; rr = NestList[RotateRight, bits, Length[bits]-1] // Rest; AllTrue[rr, FromDigits[#, 2] < n&]];
PROG
(Python)
def ok(n):
b = bin(n)[2:]
return all(b[i:] + b[:i] < b for i in range(1, len(b)))
CROSSREFS
Numbers whose binary expansion is aperiodic are A328594. Numbers whose reversed binary expansion is a necklace are A328595. Length of Lyndon factorization of binary expansion is A211100. Length of co-Lyndon factorization of binary expansion is A329312. Length of Lyndon factorization of reversed binary expansion is A329313. Length of co-Lyndon factorization of reversed binary expansion is A329326. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Lyndon compositions are A275692 (this sequence). - Co-Lyndon compositions are A326774. - Co-Lyndon factorizations are counted by A333765. - Lyndon factorizations are counted by A333940.
Number of factors in Lyndon factorization of binary expansion of n.
+10
48
1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 4, 3, 4, 4, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5, 4, 6, 5, 6, 6, 7, 2, 3, 2, 4, 2, 3, 2, 5, 3, 3, 2, 4, 2, 3, 2, 6, 3, 4, 3, 5, 4, 3, 2, 5, 3, 4, 3, 4, 3, 3, 2, 7, 3, 4, 3, 5, 3, 4, 3, 6, 4, 5, 3, 5, 4, 4, 3, 7, 4, 5, 4, 6, 5, 5, 4, 7
COMMENTS
Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). a(n) = number of factors in Lyndon factorization of binary expansion of n.
It appears that a(n) = k for the first time when n = 2^(k-1)+1.
We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. - Gus Wiseman, Nov 12 2019
REFERENCES
M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42
EXAMPLE
n=25 has binary expansion 11001, which has Lyndon factorization (1)(1)(001) with three factors, so a(25) = 3.
Here are the Lyndon factorizations for small values of n:
.0.
.1.
.1.0.
.1.1.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.1.0.0.0.
.1.001.
.1.01.0.
.1.011.
.1.1.0.0.
...
MATHEMATICA
lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #]]&]]]];
Table[Length[lynfac[IntegerDigits[n, 2]]], {n, 0, 30}] (* Gus Wiseman, Nov 12 2019 *)
CROSSREFS
Cf. A001037 (number of Lyndon words of length m); A102659 (list thereof). Ignoring the first digit gives A211097.
Length of the co-Lyndon factorization of the binary expansion of n.
+10
43
1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 2, 3, 2, 5, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3
COMMENTS
The co-Lyndon product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (231) and (213) is (212313), the product of (221) and (213) is (212213), and the product of (122) and (2121) is (1212122). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1001) has sorted co-Lyndon factorization (1)(100).
Also the length of the Lyndon factorization of the inverted binary expansion of n, where the inverted digits are 1 minus the binary digits.
EXAMPLE
The binary indices of 1..20 together with their co-Lyndon factorizations are:
1: (1) = (1)
2: (10) = (10)
3: (11) = (1)(1)
4: (100) = (100)
5: (101) = (10)(1)
6: (110) = (110)
7: (111) = (1)(1)(1)
8: (1000) = (1000)
9: (1001) = (100)(1)
10: (1010) = (10)(10)
11: (1011) = (10)(1)(1)
12: (1100) = (1100)
13: (1101) = (110)(1)
14: (1110) = (1110)
15: (1111) = (1)(1)(1)(1)
16: (10000) = (10000)
17: (10001) = (1000)(1)
18: (10010) = (100)(10)
19: (10011) = (100)(1)(1)
20: (10100) = (10100)
MATHEMATICA
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Table[Length[colynfac[IntegerDigits[n, 2]]], {n, 100}]
CROSSREFS
Cf. A000031, A001037, A059966, A060223, A211097, A296372, A296658, A329131, A329314, A329318, A329324, A329325.
Length of the Lyndon factorization of the reversed binary expansion of n.
+10
34
0, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 4, 1, 2, 1, 3, 1, 2, 1, 5, 1, 2, 2, 3, 2, 3, 2, 4, 1, 2, 3, 4, 1, 3, 2, 5, 1, 2, 2, 3, 1, 2, 2, 4, 1, 2, 1, 3, 1, 2, 1, 6, 1, 2, 2, 3, 2, 3, 2, 4, 1, 3, 3, 4, 2, 3, 2, 5, 1, 2, 2, 3, 1, 4, 3
COMMENTS
We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
EXAMPLE
The sequence of reversed binary expansions of the nonnegative integers together with their Lyndon factorizations begins:
0: () = ()
1: (1) = (1)
2: (01) = (01)
3: (11) = (1)(1)
4: (001) = (001)
5: (101) = (1)(01)
6: (011) = (011)
7: (111) = (1)(1)(1)
8: (0001) = (0001)
9: (1001) = (1)(001)
10: (0101) = (01)(01)
11: (1101) = (1)(1)(01)
12: (0011) = (0011)
13: (1011) = (1)(011)
14: (0111) = (0111)
15: (1111) = (1)(1)(1)(1)
16: (00001) = (00001)
17: (10001) = (1)(0001)
18: (01001) = (01)(001)
19: (11001) = (1)(1)(001)
20: (00101) = (00101)
MATHEMATICA
lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #1]]&]]]];
Table[If[n==0, 0, Length[lynfac[Reverse[IntegerDigits[n, 2]]]]], {n, 0, 30}]
CROSSREFS
The non-reversed version is A211100. Numbers whose reversed binary expansion is a necklace are A328595. Numbers whose reversed binary expansion is a aperiodic are A328594. Length of the co-Lyndon factorization of the binary expansion is A329312. Cf. A000031, A027375, A059966, A060223, A121016, A211097, A275692, A329131, A329314, A329317, A329325.
For any number m, let m* be the bi-infinite string obtained by repetition of the binary representation of m; this sequence lists the numbers n such that for any k < n, n* does not equal k* up to a shift.
+10
26
0, 1, 2, 4, 5, 8, 9, 11, 16, 17, 18, 19, 21, 23, 32, 33, 34, 35, 37, 38, 39, 43, 47, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 128, 129, 130, 131, 132, 133, 134, 135, 137, 138, 139, 140, 141, 142, 143, 146, 147, 149, 150, 151, 154
COMMENTS
This sequence contains every power of 2.
For any k > 0, there are A001037(k) terms with binary length k. Also numbers k such that the k-th composition in standard order (row k of A066099) is a co-Lyndon word (regular Lyndon words being A275692). For example, the sequence of all co-Lyndon words begins: 0: () 37: (3,2,1) 79: (3,1,1,1,1)
1: (1) 38: (3,1,2) 85: (2,2,2,1)
2: (2) 39: (3,1,1,1) 87: (2,2,1,1,1)
4: (3) 43: (2,2,1,1) 91: (2,1,2,1,1)
5: (2,1) 47: (2,1,1,1,1) 95: (2,1,1,1,1,1)
8: (4) 64: (7) 128: (8)
9: (3,1) 65: (6,1) 129: (7,1)
11: (2,1,1) 66: (5,2) 130: (6,2)
16: (5) 67: (5,1,1) 131: (6,1,1)
17: (4,1) 68: (4,3) 132: (5,3)
18: (3,2) 69: (4,2,1) 133: (5,2,1)
19: (3,1,1) 70: (4,1,2) 134: (5,1,2)
21: (2,2,1) 71: (4,1,1,1) 135: (5,1,1,1)
23: (2,1,1,1) 73: (3,3,1) 137: (4,3,1)
32: (6) 74: (3,2,2) 138: (4,2,2)
33: (5,1) 75: (3,2,1,1) 139: (4,2,1,1)
34: (4,2) 77: (3,1,2,1) 140: (4,1,3)
35: (4,1,1) 78: (3,1,1,2) 141: (4,1,2,1)
(End)
EXAMPLE
3* = ...11... equals 1* = ...1..., so 3 is not a term.
6* = ...110... equals up to a shift 5* = ...101..., so 6 is not a term.
11* = ...1011... only equals up to a shift 13* = ...1101... and 14* = ...1110..., so 11 is a term.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
Select[Range[0, 100], colynQ[stc[#]]&] (* Gus Wiseman, Apr 19 2020 *)
PROG
(PARI) See Links section.
CROSSREFS
Necklace compositions are counted by A008965. Lyndon compositions are counted by A059966. Length of Lyndon factorization of binary expansion is A211100. Numbers whose reversed binary expansion is a necklace are A328595. Length of co-Lyndon factorization of binary expansion is A329312. Length of Lyndon factorization of reversed binary expansion is A329313. Length of co-Lyndon factorization of reversed binary expansion is A329326. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774 (this sequence). - Aperiodic compositions are A328594. - Reversed co-necklaces are A328595. - Co-Lyndon factorizations are counted by A333765. - Lyndon factorizations are counted by A333940. - Length of co-Lyndon factorization is A334029.
Rotational period of the k-th composition in standard order; a(0) = 0.
+10
21
0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 1, 2, 2, 3, 1, 3, 3, 4, 2, 3, 1, 4, 3, 2, 4, 5, 2, 3, 3, 4, 3, 4, 2, 5, 3, 4, 4, 5, 4, 5, 5, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5, 2, 3, 3, 4, 3, 4, 4
COMMENTS
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
EXAMPLE
The a(299) = 5 rotations:
(1,1,3,2,2)
(1,3,2,2,1)
(3,2,2,1,1)
(2,2,1,1,3)
(2,1,1,3,2)
The a(9933) = 4 rotations:
(1,2,1,3,1,2,1,3)
(1,3,1,2,1,3,1,2)
(2,1,3,1,2,1,3,1)
(3,1,2,1,3,1,2,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Array[RotateRight[stc[n], #]&, DigitCount[n, 2, 1]]]], {n, 0, 100}]
CROSSREFS
Aperiodic compositions are counted by A000740. Aperiodic binary words are counted by A027375. The orderless period of prime indices is A052409. Numbers whose binary expansion is periodic are A121016. Periodic compositions are counted by A178472. The version for binary expansion is A302291. Numbers whose prime signature is aperiodic are A329139. Compositions by number of distinct rotations are A333941. All of the following pertain to compositions in standard order ( A066099): - Equal runs are counted by A124767. - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Rotational period is A333632 (this sequence).
Number of factors in Lyndon factorization of binary vectors of lengths 1, 2, 3, ...
+10
18
1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 3, 2, 3, 3, 4, 1, 2, 1, 3, 2, 2, 1, 4, 2, 3, 2, 4, 3, 4, 4, 5, 1, 2, 1, 3, 1, 2, 1, 4, 2, 3, 1, 3, 2, 2, 1, 5, 2, 3, 2, 4, 3, 3, 2, 5, 3, 4, 3, 5, 4, 5, 5, 6, 1, 2, 1, 3, 1, 2, 1, 4, 2, 2, 1, 3, 1, 2, 1, 5, 2, 3, 2, 4, 3, 2, 1, 4, 2, 3, 2, 3, 2, 2, 1, 6, 2, 3, 2, 4, 2, 3, 2, 5, 3, 4, 2, 4, 3, 3, 2, 6, 3, 4, 3, 5, 4, 4, 3, 6, 4, 5
COMMENTS
Any binary word has a unique factorization as a product of nonincreasing Lyndon words (see Lothaire). Here we look at the Lyndon factorizations of the binary vectors 0,1, 00,01,10,11, 000,001,010,011,100,101,110,111, 0000,...
The smallest (or rightmost) factors are given by A211095 and A211096, offset by 2.
REFERENCES
M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983. See Theorem 5.1.5, p. 67.
G. Melançon, Factorizing infinite words using Maple, MapleTech Journal, vol. 4, no. 1, 1997, pp. 34-42
EXAMPLE
Here are the Lyndon factorizations of the first few binary vectors:
.0.
.1.
.0.0.
.01.
.1.0.
.1.1.
.0.0.0.
.001.
.01.0. <- this means that the factorization is (01)(0), for example
.011.
.1.0.0.
.1.01.
.1.1.0.
.1.1.1.
.0.0.0.0.
...
MATHEMATICA
lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #]]&]]]];
Table[Length[lynfac[Rest[IntegerDigits[n, 2]]]], {n, 2, 50}] (* Gus Wiseman, Nov 14 2019 *)
CROSSREFS
A211098 and A211099 give information about the largest (or leftmost) factor.
Retaining the first digit gives A211100. Binary Lyndon words are counted by A001037 and constructed by A102659. Numbers whose reversed binary expansion is Lyndon are A328596.
Number of rotational symmetries in the binary expansion of a number.
+10
17
1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
COMMENTS
Mersenne numbers of form (2^n - 1) have n rotational symmetries.
For prime length binary expansions these are the only nontrivial symmetries.
For composite length expansions it seems that when the number of symmetries is nontrivial it is equal to a factor of the length. We're working on an explicit formula.
Discovered in the context of random circulant matrices, examining if there's a correlation between degrees of freedom and number of symmetries in the first row.
When combined with A138954, these two sequences should give a full account of the number of redundant rows in a circulant square matrix with at most two distinct values, where a(n) is the encoding of the first row of the matrix into binary such that value a = 1 and value b = 0. Discovered on the night of Apr 02, 2008 by Maxwell Sills and Gary Doran.
Conjecture: For binary expansions of length n, there are d(n) distinct values that will show up as symmetries, where d is the divisor function. The symmetry values will be precisely the divisors of n.
Example: for binary expansions of length 12, one sees that d(12) = 6 distinct values show up as symmetries (1, 2, 3, 4, 6, 12).
Conjecture: For numbers whose binary expansion has length n which has proper divisors which are all coprime: There will be only one number of length n with n symmetries. That number is 2^n - 1. For each proper divisor d (excluding 1), you can generate all numbers of length n that have n/d symmetries like so: (2^0 + 2^d + 2^2d ... 2^(n-d)) * a, where 2^(d-1) <= a < (2^d) - 1. The rest of the expansions of length n will have only the trivial symmetry.
Also the number of rotational symmetries of the n-th composition in standard order (graded reverse-lexicographic). This composition (row n of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of n, prepending 0, taking first differences, and reversing again. - Gus Wiseman, Apr 19 2020 Aperiodic compositions are counted by A000740. Aperiodic binary words are counted by A027375. The orderless period of prime indices is A052409. Numbers whose binary expansion is periodic are A121016. Periodic compositions are counted by A178472. Period of binary expansion is A302291. Compositions by sum and number of distinct rotations are A333941. All of the following pertain to compositions in standard order ( A066099): - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Reversed co-necklaces are A328595. (End).
EXAMPLE
a(10) = 2 because the binary expansion of 10 is 1010 and it has two rotational symmetries (including identity).
MATHEMATICA
Table[IntegerLength[n, 2]/Length[Union[Array[RotateRight[IntegerDigits[n, 2], #]&, IntegerLength[n, 2]]]], {n, 100}] (* Gus Wiseman, Apr 19 2020 *)
Numbers whose binary expansion without the most significant (first) digit has Lyndon and co-Lyndon factorizations of equal lengths.
+10
17
1, 2, 3, 4, 7, 8, 10, 13, 15, 16, 22, 25, 31, 32, 36, 42, 46, 49, 53, 59, 63, 64, 76, 82, 94, 97, 109, 115, 127, 128, 136, 148, 156, 162, 166, 169, 170, 172, 181, 182, 190, 193, 201, 202, 211, 213, 214, 217, 221, 227, 235, 247, 255, 256, 280, 292, 306, 308
COMMENTS
We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon word is a finite sequence that is prime with respect to the Lyndon product. Equivalently, a Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into Lyndon words, and if these factors are arranged in lexicographically decreasing order, their concatenation is equal to their Lyndon product. For example, (1001) has sorted Lyndon factorization (001)(1).
Similarly, the co-Lyndon product is the lexicographically minimal sequence obtainable by shuffling the sequences together, and a co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product, or, equivalently, a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. For example, (1001) has sorted co-Lyndon factorization (1)(100).
EXAMPLE
The sequence of terms together with their trimmed binary expansions and their co-Lyndon and Lyndon factorizations begins:
1: () = 0 = 0
2: (0) = (0) = (0)
3: (1) = (1) = (1)
4: (00) = (0)(0) = (0)(0)
7: (11) = (1)(1) = (1)(1)
8: (000) = (0)(0)(0) = (0)(0)(0)
10: (010) = (0)(10) = (01)(0)
13: (101) = (10)(1) = (1)(01)
15: (111) = (1)(1)(1) = (1)(1)(1)
16: (0000) = (0)(0)(0)(0) = (0)(0)(0)(0)
22: (0110) = (0)(110) = (011)(0)
25: (1001) = (100)(1) = (1)(001)
31: (1111) = (1)(1)(1)(1) = (1)(1)(1)(1)
32: (00000) = (0)(0)(0)(0)(0) = (0)(0)(0)(0)(0)
36: (00100) = (0)(0)(100) = (001)(0)(0)
42: (01010) = (0)(10)(10) = (01)(01)(0)
46: (01110) = (0)(1110) = (0111)(0)
49: (10001) = (1000)(1) = (1)(0001)
53: (10101) = (10)(10)(1) = (1)(01)(01)
59: (11011) = (110)(1)(1) = (1)(1)(011)
63: (11111) = (1)(1)(1)(1)(1) = (1)(1)(1)(1)(1)
MATHEMATICA
lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
lynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[lynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], lynQ[Take[q, #]]&]]]];
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]]@Last[Select[Range[Length[q]], colynQ[Take[q, #]]&]]];
Select[Range[100], Length[lynfac[Rest[IntegerDigits[#, 2]]]]==Length[colynfac[Rest[IntegerDigits[#, 2]]]]&]
CROSSREFS
Lyndon and co-Lyndon compositions are (both) counted by A059966. Numbers whose reversed binary expansion is Lyndon are A328596. Numbers whose binary expansion is co-Lyndon are A275692. Cf. A001037, A060223, A102659, A211100, A329131, A329312, A329313, A329318, A329326, A329394, A329398.
Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.
+10
16
1, 2, 2, 1, 3, 2, 1, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 3, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 2, 3, 2, 1, 4, 1, 1, 4, 2, 5, 1, 6, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 2, 3, 3, 1, 4, 1, 1, 1, 4, 1, 2, 4, 2, 1, 4, 3, 5, 1, 1, 5, 2, 6, 1, 7
COMMENTS
The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling them together. Every finite positive integer sequence has a unique *-factorization using prime compositions P = {(1), (2), (21), (3), (211), ...}. See A060223 and A228369 for details. These are co-Lyndon compositions, ordered first by sum and then lexicographically. - Gus Wiseman, Nov 15 2019
FORMULA
Row lengths are A059966(n) = number of prime compositions of n.
EXAMPLE
The prime factorization of (1, 1, 4, 2, 3, 1, 5, 5) is: (11423155) = (1)*(1)*(5)*(5)*(4231). The prime factorizations of the initial terms of A000002 are: (1) = (1)
(12) = (1)*(2)
(122) = (1)*(2)*(2)
(1221) = (1)*(221)
(12211) = (1)*(2211)
(122112) = (1)*(2)*(2211)
(1221121) = (1)*(221121)
(12211212) = (1)*(2)*(221121)
(122112122) = (1)*(2)*(2)*(221121)
(1221121221) = (1)*(221)*(221121)
(12211212212) = (1)*(2)*(221)*(221121)
(122112122122) = (1)*(2)*(2)*(221)*(221121).
Read as a sequence:
(1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
Read as a triangle:
(1)
(2)
(21), (3)
(211), (31), (4)
(2111), (221), (311), (32), (41), (5).
Read as a sequence of triangles:
1 2 2 1 2 1 1 2 1 1 1 2 1 1 1 1 2 1 1 1 1 1
3 3 1 2 2 1 2 2 1 1 2 1 2 1 1
4 3 1 1 3 1 1 1 2 2 1 1 1
3 2 3 1 2 2 2 2 1
4 1 3 2 1 3 1 1 1 1
5 4 1 1 3 1 1 2
4 2 3 1 2 1
5 1 3 2 1 1
6 3 2 2
3 3 1
4 1 1 1
4 1 2
4 2 1
4 3
5 1 1
5 2
6 1
7.
MATHEMATICA
colynQ[q_]:=Array[Union[{RotateRight[q, #], q}]=={RotateRight[q, #], q}&, Length[q]-1, 1, And];
lexsort[f_, c_]:=OrderedQ[PadRight[{f, c}]];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n], colynQ], lexsort], {n, 5}] (* Gus Wiseman, Nov 15 2019 *)
CROSSREFS
The binary non-"co" version is A102659. A sequence listing all Lyndon compositions is A294859. Numbers whose binary expansion is co-Lyndon are A328596. Numbers whose binary expansion is co-Lyndon are A275692. Cf. A211097, A211100, A296372, A296373, A298941, A329131, A329312, A329313, A329314, A329324, A329326.
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