![[Information Index]](ico/objinfo.gif)
![[Generation Index]](ico/objgen.gif)
![[COS homepage]](ico/cos.gif)
6-ary Lyndon words with given trace and subtrace.
A 6-ary Lyndon word is a string made from the characters
0, 1, 2, 3, 4, 5.
It must be aperiodic (not equal to any of its non-trivial rotations) and
be lexicographically least among its rotations.
Here we consider the set L(n;t,s) of length
n
lyndon words
a1a2···an
over the alphabet {0,1,2,3,4,5} that have trace t
and subtrace s. The trace of a 6-ary lyndon word
is the sum of its digits mod 6, i.e.
t = a1+a2+ ··· +an
mod 6. The subtrace is the sum of the products
of all n(n-1)/2 pairs of digits taken mod 6, i.e.
s = SUM( aiaj :
1 < i < j < n ).
|
| (trace,subtrace) |
| n
| (0,0)
| (0,1)
| (0,2)
| (0,3)
| (0,4)
| (0,5)
| (1,0)
(5,0)
| (1,1)
(5,1)
| (1,2)
(5,2)
| (1,3)
(5,3)
| (1,4)
(5,4)
| (1,5)
(5,5)
| (2,0)
(4,0)
| (2,1)
(4,1)
| (2,2)
(4,2)
| (2,3)
(4,3)
| (2,4)
(4,4)
| (2,5)
(4,5)
| (3,0)
| (3,1)
| (3,2)
| (3,3)
| (3,4)
| (3,5)
|
| 1 |
1 | 0 | 0
| 0 | 0 | 0
| 1 | 0 | 0
| 0 | 0 | 0
| 1 | 0 | 0
| 0 | 0 | 0
| 1 | 0 | 0
| 0 | 0 | 0
|
|---|
| 2 |
0 | 0 | 1
| 0 | 0 | 1
| 2 | 0 | 0
| 0 | 1 | 0
| 1 | 0 | 0
| 1 | 0 | 0
| 1 | 0 | 2
| 0 | 0 | 0
|
|---|
| 3 |
0 | 0 | 2
| 3 | 0 | 6
| 3 | 1 | 3
| 1 | 3 | 1
| 1 | 3 | 1
| 3 | 1 | 3
| 3 | 0 | 6
| 0 | 0 | 2
|
|---|
| 4 |
4 | 8 | 6
| 13 | 2 | 18
| 9 | 6 | 12
| 9 | 6 | 12
| 4 | 8 | 6
| 13 | 2 | 18
| 9 | 6 | 12
| 9 | 6 | 12
|
|---|
| 5 |
25 | 60 | 36
| 42 | 36 | 60
| 36 | 42 | 36
| 60 | 25 | 60
| 36 | 42 | 36
| 60 | 25 | 60
| 25 | 60 | 36
| 42 | 36 | 60
|
|---|
| 6 |
165 | 236 | 240
| 165 | 236 | 240
| 162 | 270 | 162
| 270 | 162 | 270
| 214 | 214 | 214
| 214 | 214 | 214
| 126 | 300 | 180
| 207 | 180 | 300
|
|---|
| 7 |
1157 | 1008 | 1296
| 900 | 1296 | 1008
| 900 | 1296 | 1008
| 1157 | 1008 | 1296
| 1157 | 1008 | 1296
| 900 | 1296 | 1008
| 900 | 1296 | 1008
| 1157 | 1008 | 1296
|
|---|
| 8 |
6552 | 5280 | 6312
| 5094 | 6792 | 4908
| 5616 | 5832 | 6048
| 5616 | 5832 | 6048
| 6312 | 5094 | 6792
| 4908 | 6552 | 5280
| 5832 | 6048 | 5616
| 5832 | 6048 | 5616
|
|---|
| 9 |
33045 | 30240 | 31824
| 29151 | 34272 | 28080
| 33048 | 29160 | 33048
| 29160 | 33048 | 29160
| 33048 | 29160 | 33048
| 29160 | 33048 | 29160
| 33045 | 30240 | 31824
| 29151 | 34272 | 28080
|
|---|
| 10 |
167928 | 169987 | 165840
| 167928 | 169987 | 165840
| 178459 | 159408 | 176256
| 157464 | 180662 | 155520
| 167928 | 169987 | 165840
| 167928 | 169987 | 165840
| 178459 | 159408 | 176256
| 157464 | 180662 | 155520
|
|---|
Examples:
-
The three 6-ary Lyndon words of trace 2, subtrace 1 and length
3 are ( 011, 134, 143 }.
-
The two 6-ary Lyndon words of trace 3, subtrace 5 and length
3 are { 135, 153 }.
-
The six 6-ary Lyndon words of trace 5, subtrace 1 and length
4 are { 0113, 0131, 0311, 1334, 1343, 1433 }.
Further Notes:
-
Column (0,0) is sequence
A074422 in
Neil J. Sloane's
database
of integer sequences.
-
Column (0,1) is sequence
A074423 in
Neil J. Sloane's
database
of integer sequences.
-
Column (0,2) is sequence
A074424 in
Neil J. Sloane's
database
of integer sequences.
-
Column (0,3) is sequence
A074425 in
Neil J. Sloane's
database
of integer sequences.
-
Column (0,4) is sequence
A074426 in
Neil J. Sloane's
database
of integer sequences.
-
Column (0,5) is sequence
A074427 in
Neil J. Sloane's
database
of integer sequences.
-
Column (1,0),(5,0) is sequence
A074428 in
Neil J. Sloane's
database
of integer sequences.
-
Column (1,1),(5,1) is sequence
A074429 in
Neil J. Sloane's
database
of integer sequences.
-
Column (1,2),(5,2) is sequence
A074430 in
Neil J. Sloane's
database
of integer sequences.
-
Column (1,3),(5,3) is sequence
A074431 in
Neil J. Sloane's
database
of integer sequences.
-
Column (1,4),(5,4) is sequence
A074432 in
Neil J. Sloane's
database
of integer sequences.
-
Column (1,5),(5,5) is sequence
A074433 in
Neil J. Sloane's
database
of integer sequences.
-
Column (2,0),(4,0) is sequence
A074434 in
Neil J. Sloane's
database
of integer sequences.
-
Column (2,1),(4,1) is sequence
A074435 in
Neil J. Sloane's
database
of integer sequences.
-
Column (2,2),(4,2) is sequence
A074436 in
Neil J. Sloane's
database
of integer sequences.
-
Column (2,3),(4,3) is sequence
A074437 in
Neil J. Sloane's
database
of integer sequences.
-
Column (2,4),(4,4) is sequence
A074438 in
Neil J. Sloane's
database
of integer sequences.
-
Column (2,5),(4,5) is sequence
A074439 in
Neil J. Sloane's
database
of integer sequences.
-
Column (3,0) is sequence
A074440 in
Neil J. Sloane's
database
of integer sequences.
-
Column (3,1) is sequence
A074441 in
Neil J. Sloane's
database
of integer sequences.
-
Column (3,2) is sequence
A074442 in
Neil J. Sloane's
database
of integer sequences.
-
Column (3,3) is sequence
A074443 in
Neil J. Sloane's
database
of integer sequences.
-
Column (3,4) is sequence
A074444 in
Neil J. Sloane's
database
of integer sequences.
-
Column (3,5) is sequence
A074445 in
Neil J. Sloane's
database
of integer sequences.
Questions?? Email
The wizard of COS.
(Please note that the suffix XXXX must be removed from the preceeding
email address.)
It was last updated Wednesday, 10-May-2006 10:32:13 PDT.
There have been 1012 visitors to this page since May 16, 2000
.
©Frank Ruskey, 1995-2003.