Mixing
Does This Property of Words Have a Name?
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Let us say an infinite word $w=w_1w_2cdots$ over a finite alphabet ${a_1,ldots,a_r}$ is good if there exists a positive integer $m$ such that none of the words $a_1^m,ldots,a_r^m$ appear as factors of $w$. Equivalently, saying $w$ is good means that it does not contain arbitrarily long factors that use only one letter. Does this "good" property already have a different name in the literature?
reference-request terminology definition combinatorics-on-words
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
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add a comment |
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Let us say an infinite word $w=w_1w_2cdots$ over a finite alphabet ${a_1,ldots,a_r}$ is good if there exists a positive integer $m$ such that none of the words $a_1^m,ldots,a_r^m$ appear as factors of $w$. Equivalently, saying $w$ is good means that it does not contain arbitrarily long factors that use only one letter. Does this "good" property already have a different name in the literature?
reference-request terminology definition combinatorics-on-words
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
$endgroup$
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It's an example of "shift of finite type" . But there are other examples not of this form.
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– kimchi lover
Jan 28 at 23:42
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As mentioned, the set of sequences $X_{mathcal{F}}$ such that no word in the finite set $mathcal{F} = {a_i^m mid 1 leq i leq r}$ appears in any sequence in $X_{mathcal{F}}$ is called a shift of finite type (SFT) or topological Markov shift. You might also be interested in the related notion of square-free and $k$-free sequences (not quite the same, but of the same spirit).
$endgroup$
– Dan Rust
Jan 28 at 23:48
add a comment |
$begingroup$
Let us say an infinite word $w=w_1w_2cdots$ over a finite alphabet ${a_1,ldots,a_r}$ is good if there exists a positive integer $m$ such that none of the words $a_1^m,ldots,a_r^m$ appear as factors of $w$. Equivalently, saying $w$ is good means that it does not contain arbitrarily long factors that use only one letter. Does this "good" property already have a different name in the literature?
reference-request terminology definition combinatorics-on-words
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
$endgroup$
Let us say an infinite word $w=w_1w_2cdots$ over a finite alphabet ${a_1,ldots,a_r}$ is good if there exists a positive integer $m$ such that none of the words $a_1^m,ldots,a_r^m$ appear as factors of $w$. Equivalently, saying $w$ is good means that it does not contain arbitrarily long factors that use only one letter. Does this "good" property already have a different name in the literature?
reference-request terminology definition combinatorics-on-words
reference-request terminology definition combinatorics-on-words
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
asked Jan 28 at 23:40
Colin DefantColin Defant
712312
712312
$begingroup$
It's an example of "shift of finite type" . But there are other examples not of this form.
$endgroup$
– kimchi lover
Jan 28 at 23:42
$begingroup$
As mentioned, the set of sequences $X_{mathcal{F}}$ such that no word in the finite set $mathcal{F} = {a_i^m mid 1 leq i leq r}$ appears in any sequence in $X_{mathcal{F}}$ is called a shift of finite type (SFT) or topological Markov shift. You might also be interested in the related notion of square-free and $k$-free sequences (not quite the same, but of the same spirit).
$endgroup$
– Dan Rust
Jan 28 at 23:48
add a comment |
$begingroup$
It's an example of "shift of finite type" . But there are other examples not of this form.
$endgroup$
– kimchi lover
Jan 28 at 23:42
$begingroup$
As mentioned, the set of sequences $X_{mathcal{F}}$ such that no word in the finite set $mathcal{F} = {a_i^m mid 1 leq i leq r}$ appears in any sequence in $X_{mathcal{F}}$ is called a shift of finite type (SFT) or topological Markov shift. You might also be interested in the related notion of square-free and $k$-free sequences (not quite the same, but of the same spirit).
$endgroup$
– Dan Rust
Jan 28 at 23:48
$begingroup$
It's an example of "shift of finite type" . But there are other examples not of this form.
$endgroup$
– kimchi lover
Jan 28 at 23:42
$begingroup$
It's an example of "shift of finite type" . But there are other examples not of this form.
$endgroup$
– kimchi lover
Jan 28 at 23:42
$begingroup$
As mentioned, the set of sequences $X_{mathcal{F}}$ such that no word in the finite set $mathcal{F} = {a_i^m mid 1 leq i leq r}$ appears in any sequence in $X_{mathcal{F}}$ is called a shift of finite type (SFT) or topological Markov shift. You might also be interested in the related notion of square-free and $k$-free sequences (not quite the same, but of the same spirit).
$endgroup$
– Dan Rust
Jan 28 at 23:48
$begingroup$
As mentioned, the set of sequences $X_{mathcal{F}}$ such that no word in the finite set $mathcal{F} = {a_i^m mid 1 leq i leq r}$ appears in any sequence in $X_{mathcal{F}}$ is called a shift of finite type (SFT) or topological Markov shift. You might also be interested in the related notion of square-free and $k$-free sequences (not quite the same, but of the same spirit).
$endgroup$
– Dan Rust
Jan 28 at 23:48
add a comment |
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$begingroup$
It's an example of "shift of finite type" . But there are other examples not of this form.
$endgroup$
– kimchi lover
Jan 28 at 23:42
$begingroup$
As mentioned, the set of sequences $X_{mathcal{F}}$ such that no word in the finite set $mathcal{F} = {a_i^m mid 1 leq i leq r}$ appears in any sequence in $X_{mathcal{F}}$ is called a shift of finite type (SFT) or topological Markov shift. You might also be interested in the related notion of square-free and $k$-free sequences (not quite the same, but of the same spirit).
$endgroup$
– Dan Rust
Jan 28 at 23:48