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JMP
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For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points. In general, a mathematical support means as an assistance to the main act.

To try to define the term, we can first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial simple transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of simple transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of simple transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying simple transposition sets for a permutation $\pi$, or perhaps, the minimum number of simple transpositions needed to return $\pi$ to $S$.

The actual statistic can be easily calculated by creating an array of $n-1$, and ticking every cell that is crossed by returning an element in a straight path to it's original position, and counting the number of ticks, max 1 per cell.

For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points. In general, a mathematical support means as an assistance to the main act.

To try to define the term, we can first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial simple transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of simple transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of simple transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying simple transposition sets for a permutation $\pi$, or perhaps, the minimum number of simple transpositions needed to return $\pi$ to $S$.

For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points. In general, a mathematical support means as an assistance to the main act.

To try to define the term, we can first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial simple transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of simple transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of simple transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying simple transposition sets for a permutation $\pi$, or perhaps, the minimum number of simple transpositions needed to return $\pi$ to $S$.

The actual statistic can be easily calculated by creating an array of $n-1$, and ticking every cell that is crossed by returning an element in a straight path to it's original position, and counting the number of ticks, max 1 per cell.

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JMP
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For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points. In general, a mathematical support means as an assistance to the main act.

To try to define the term, we can first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial simple transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of simple transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of simple transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying simple transposition sets for a permutation $\pi$, or perhaps, the minimum number of simple transpositions needed to return $\pi$ to $S$.

For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points.

To define the term, first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying transposition sets for a permutation $\pi$.

For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points. In general, a mathematical support means as an assistance to the main act.

To try to define the term, we can first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial simple transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of simple transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of simple transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying simple transposition sets for a permutation $\pi$, or perhaps, the minimum number of simple transpositions needed to return $\pi$ to $S$.

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Michael Hardy
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For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points.

To define the term, first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\large\cup_t \;_\pi U_n(t)$$$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying transposition sets for a permutation $\pi$.

For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points.

To define the term, first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\large\cup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying transposition sets for a permutation $\pi$.

For the first part of your question, support doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points.

To define the term, first define the process that arrives at the statistic:

Let $T_n$ be the set of all non-trivial transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;_\pi T_n=\left\{\{t_1,\dots t_k\}\in T_n^k:\pi\prod_\limits{i=1}^k t_i=S\right\}$$

i.e. $\;_\pi T_n$ is the set of sets of transpositions that take $\pi$ to the identity $S$.

Define:

$$\;_\pi U_n(t\in \;_\pi T_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$_\pi U_n=\bigcup_t \;_\pi U_n(t)$$

as the set of sets that contain the unique elements of transposition sets from $\;_\pi T_n$.

Define $\;_\pi V_n=\{|v|:v\in \;_\pi U_n\}$ and then the statistic we want is $\min(\;_\pi V_n)$.

So, minimum cardinality of the identifying transposition sets for a permutation $\pi$.

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