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Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests the following unsurprising result:

There is a $k=k(m)$ integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

The conjecture from the question is that $a_1$ is about $\frac{m}2$, $a_2$ is about $\frac{m}4$ and so on. That is a bit vague. Does it mean that $a_j$ is about $\frac{m}{2^j}$ (perhaps for $1 \leq j \ll k?.$ An now deleted answer speculated that $a_1=2a_2=3a_3=\cdots=ja_j$ again perhaps for $1 \leq j \ll k.$

The present calculations are not sufficient to test if either of these is likely.

• For $m=1$ the maximum occurs for $1\ (n-1)$
• For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
• For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
• For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
• For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
• For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ except that
• for $n=22$ there is a tie between $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
• For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search.

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

• $1^8$
• $1^72$
• $1^62^2$
• $1^62\,3$
• $1^52^23$
• $1^42^33$
• $1^5\,2\,3\,4$
• $1^42^23\,4$
• $1^42^23\,5$
• For $n=17$ a tie between $1^42^23\,6$ and $1^32^23^24$
• $1^4\,2\,3\,4\,5$ for $n=18.$
• $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
• for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$

Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests the following unsurprising result:

There is a $k=k(m)$ integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

later

Based on the calculations one might speculate further that the $a_i$ decrease until they are $1$ (perhaps no later than $\sqrt{k}$ or some such $j \ll k$)

The conjecture from the question is that $a_1$ is about $\frac{m}2$, $a_2$ is about $\frac{m}4$ and so on. That is a bit vague. Does it mean that $a_j$ is about $\frac{m}{2^j}$ (perhaps for $1 \leq j \ll k?.$ An now deleted answer speculated that $a_1=2a_2=3a_3=\cdots=ja_j$ again perhaps for $1 \leq j \ll k.$

The present calculations are not sufficient to test if either of these is likely.

• For $m=1$ the maximum occurs for $1\ (n-1)$
• For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
• For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
• For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
• For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
• For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ except that
• for $n=22$ there is a tie between $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
• For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search.

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

• $1^8$
• $1^72$
• $1^62^2$
• $1^62\,3$
• $1^52^23$
• $1^42^33$
• $1^5\,2\,3\,4$
• $1^42^23\,4$
• $1^42^23\,5$
• For $n=17$ a tie between $1^42^23\,6$ and $1^32^23^24$
• $1^4\,2\,3\,4\,5$ for $n=18.$
• $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
• for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$

Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests the following unsurprising result:

There is a $k=k(m)$ integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

The particular counts do not seem to support the conjecture from the question.

• For $m=1$ the maximum occurs for $1\ (n-1)$
• For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
• For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
• For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
• For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
• For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ except that
• for $n=22$ there is a tie between $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
• For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search.

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

• $1^8$
• $1^72$
• $1^62^2$
• $1^62\,3$
• $1^52^23$
• $1^42^33$
• $1^5\,2\,3\,4$
• $1^42^23\,4$
• $1^42^23\,5$
• For $n=17$ a tie between $1^42^23\,6$ and $1^32^23^24$
• $1^4\,2\,3\,4\,5$ for $n=18.$
• $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
• for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$

Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests the following unsurprising result:

There is a $k=k(m)$ integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

The conjecture from the question is that $a_1$ is about $\frac{m}2$, $a_2$ is about $\frac{m}4$ and so on. That is a bit vague. Does it mean that $a_j$ is about $\frac{m}{2^j}$ (perhaps for $1 \leq j \ll k?.$ An now deleted answer speculated that $a_1=2a_2=3a_3=\cdots=ja_j$ again perhaps for $1 \leq j \ll k.$

The present calculations are not sufficient to test if either of these is likely.

• For $m=1$ the maximum occurs for $1\ (n-1)$
• For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
• For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
• For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
• For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
• For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ except that
• for $n=22$ there is a tie between $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
• For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search.

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

• $1^8$
• $1^72$
• $1^62^2$
• $1^62\,3$
• $1^52^23$
• $1^42^33$
• $1^5\,2\,3\,4$
• $1^42^23\,4$
• $1^42^23\,5$
• For $n=17$ a tie between $1^42^23\,6$ and $1^32^23^24$
• $1^4\,2\,3\,4\,5$ for $n=18.$
• $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
• for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$

Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests the following unsurprising result:

There is a $k=k(m)$ integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

The particular counts do not seem to support the conjecture from the question.

• For $m=1$ the maximum occurs for $1\ (n-1)$
• For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
• For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
• For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
• For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
• For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ except that
• for $n=22$ there is a tie between $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
• For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search.

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

• $1^8$
• $1^72$
• $1^62^2$
• $1^62\,3$
• $1^52^23$
• $1^42^33$
• $1^5\,2\,3\,4$
• $1^42^23\,4$
• $1^42^23\,5$
• For $n=17$ a tie between $1^42\,3\,6$ and $1^32^23^24$
• $1^4\,2\,3\,4\,5$ for $n=18.$
• $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
• for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$

Fix $m.$ For each $n$ consider the the question: among all the conjugacy classes for permutations in $S_n$ with exactly $m+1$ cycles, which has the largest size?

I will write $\prod_{i \geq 1}i^{a_i}$ to describe a partition with $a_i$ cycles of length $i.$

Below is some data. It is the result of rather unsophisticated exhaustive calculation. It strongly suggests the following unsurprising result:

There is a $k=k(m)$ integers $a_1 \geq a_2 \geq \cdots \geq a_k=1$ and $t=t(m)=\sum_1^ka_kk$ such that, for $n$ large enough relative to $m,$ the unique maximum occurs for type $1^{a_1}2^{a_2}\cdots k^{a_k}(n-t)$

The particular counts do not seem to support the conjecture from the question.

• For $m=1$ the maximum occurs for $1\ (n-1)$
• For $m=2$ and $n \gt 5$ the maximum occurs for $1\ 2\ (n-3)$
• For $m=3$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ (n-4)$
• For $m=4$ and $n \gt 10$ the maximum occurs for $1^2\ 2\ \ 3 (n-7)$
• For $m=5$ and $n \gt 20$ the maximum occurs for $1^3\ 2\ 3\ (n-8)$
• For $m=6$ and $n \gt 17$ the maximum occurs for $1^3\ 2\ 3\ 4\ (n-12)$ except that
• for $n=22$ there is a tie between $1^3\ 2\ 3\ 4\ (10)$ and $1^2\ 2\ 3\ 4\ 5\ 6$
• For $m=7$ and $n \gt 29$ the maximum occurs for $1^3\ 2^2\ 3\ 4\ (n-14)$

This might allow a more refined conjecture. Certainly data for larger $m$ would be suggestive. That would justify taking the time to do design a more intelligent search.

Here are more detailed results on $m=7$ to show exactly what I know for sure:

For $30 \leq n \leq 60$ the maximum occurs only for $1^3\ 2^2\ 3\ 4\ (n-14).$ Presumably the same is true for all $30 \leq n.$ It is also true for $n=19,20,21,22. $

the other cases are:

• $1^8$
• $1^72$
• $1^62^2$
• $1^62\,3$
• $1^52^23$
• $1^42^33$
• $1^5\,2\,3\,4$
• $1^42^23\,4$
• $1^42^23\,5$
• For $n=17$ a tie between $1^42^23\,6$ and $1^32^23^24$
• $1^4\,2\,3\,4\,5$ for $n=18.$
• $1^32\,3\,4\,5\,(n-17)$ for $23 \leq n \leq 28$
• for $n=29$ a tie between $1^3\,2\,3\,4\,5\,12$ and $1^32^23\,4\,15$