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While working on another problem, I found the following combinatorial equality, but I got it analytically, and I'm curious to find a counting argument.

Fix $n$ a positive integer. For $n_1\leq n_2\leq \cdots \leq n_k$ with $\sum n_i=n$, let $s_{n_1,\dots,n_k}$ be the number of permutations in $\Sigma_n$ with sorted cycle lengths $n_1,n_2,\dots,n_k$.

Then show:

$$\sum_{n_1,\dots,n_k} 4^{k-1}s_{n_1,\dots,n_k}=n\cdot n!$$

where the sum is restricted to the case where all $n_i$ are odd.

I suppose we could just write $t_k$ as the number of permutations in $\Sigma_n$ composted of $k$ odd cycles, and write it as $\sum_{k} 4^{k-1}t_k = n\cdot n!$.

For example, for $n=5$, the possible permutations of signature $s_{5}=4!$, $s_{1,1,3}=2\binom{5}{2}=20$, $s_{1,1,1,1,1}=1$ so the sum is:

$$4^0\cdot24+4^2\cdot 20 + 4^4\cdot 1=600=5\cdot 5!$$

I have a proof of this, which is gross - it involves substituting the power series for $\theta=\arctan x$ into the power series for $\sin 4\theta$ and $\cos 4\theta$ (for odd and even $n$, respectively.) That seems unpleasant, so I am seeking a more direct combinatorial argument.

One thing I considered was the possibility of using the identity:

$$\sum_{n=0}^{m} n\cdot n! =(m+1)!-1$$

Community
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Thomas Andrews
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2 Answers2

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Permutations with odd cycles only and cycle count marked have the species

$$\mathfrak{P}(\mathcal{U}\mathfrak{C}_{=1}(\mathcal{Z}) + \mathcal{U}\mathfrak{C}_{=3}(\mathcal{Z}) + \mathcal{U}\mathfrak{C}_{=5}(\mathcal{Z}) + \mathcal{U}\mathfrak{C}_{=7}(\mathcal{Z}) + \cdots).$$

which yields the EGF

$$G(z,u) = \exp\left(uz+ u\frac{z^3}{3}+ u\frac{z^5}{5}+ u\frac{z^7}{7}+\cdots\right) \\ = \exp\left(u\sum_{k\ge 0}\frac{z^{2k+1}}{2k+1}\right) \\ = \exp\left(u\log\frac{1}{1-z} - u\sum_{k\ge 1} \frac{z^{2k}}{2k}\right) \\ = \exp\left(u\log\frac{1}{1-z} - u\frac{1}{2}\log\frac{1}{1-z^2}\right).$$

Now here we have $u=4$ so we continue with

$$\frac{1}{(1-z)^4} \exp\left(2\log(1-z^2)\right) = \frac{(1-z^2)^2}{(1-z)^4} = \frac{(1+z)^2}{(1-z)^2} \\ = (1+2z+z^2) \frac{1}{(1-z)^2}.$$

We get as our result the quantity

$$\frac{1}{4} n! [z^n] G(z, 4) = \frac{1}{4} n! \left({n+1\choose 1} + 2{n\choose 1} + {n-1\choose 1}\right) \\ = \frac{1}{4} n! (n+1+2n+n-1) = n! \times n.$$

Marko Riedel
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  • That first step is much like my first step - I was using $z=ix$ so $\sum z^{2k+1}/(2k+1)=i\arctan x$. It's still not quite a "counting argument" as I wanted. – Thomas Andrews Apr 05 '16 at 21:01
  • I did notice that $\arctan(iz)/i$ appears here so possibly the above would not be considered a combinatorial proof. Let's hear what the combinatorial wizards have to say. – Marko Riedel Apr 05 '16 at 21:05
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    It's still a nice answer, and comes up with a more general formula with $4$ replaced by $2j$ - then the coefficient is always $n!$ times a polynomial of degree $j$ in $n$. – Thomas Andrews Apr 05 '16 at 21:39
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    If I go to the last expression prior to subbing $u=4$, it seems that $$\log\frac{1}{1-z}- \frac{1}{2}\log\frac{1}{1-z^2}=\frac{1}{2}\log\frac{1+z}{1-z}$$ and so one can simplify to $G(z,u)=\left(\frac{1+z}{1-z}\right)^{u/2}$. This also agrees with the $u=4$ case, and seems a bit simpler in statement. – Semiclassical Apr 05 '16 at 22:18
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I am going to introduce a fancy structure, that is a coloured permutation.

A coloured permutation $\sigma$ is a permutation acting on $\{1,2,\ldots,n\}$, with the property that every element has an odd order. Moreover, the elements of $\{1,\ldots,n\}$ are coloured with some colour from $\{\text{cyan},\text{magenta},\text{yellow},\text{black}\}$ and the permutation is colour-preserving, i.e. the colour of $\sigma(a)$ is the same as the colour of $a$. We want to count the number of coloured permutations, given by: $$ L_n=\!\!\!\!\sum_{\substack{n_1,\ldots,n_k \text{ odd}\\ n_1+\ldots n_k=n}}\!\!\! 4^k\cdot s_{n_1,\ldots,n_k}.$$

Preliminary lemma: the number of permutations of $\{1,\ldots,k\}$ such that every element has an odd order is given by: $$ C_k = k!\cdot [x^k]\frac{1}{1-x}\cdot\exp\left(-\frac{x^2}{2}-\frac{x^4}{4}-\ldots\right)= k!\cdot [x^k]\sqrt{\frac{1+x}{1-x}}. $$

Consequence: the number of coloured permutations is given by $$ \sum_{c+m+y+k=n}\binom{n}{c}\binom{n-c}{m}\binom{n-c-m}{y} C_c C_m C_y C_k = n!\cdot [x^n]\left(\frac{1+x}{1-x}\right)^2=4n\cdot n!.$$

Done. Double counting wins again.

Another chance would be given by showing that $L_{n+1}-L_n = (n+1)!-n!$ by some recursive argument. However, I was not able to find it.

Jack D'Aurizio
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  • This is nice work (+1) even though it uses the same EGF for the $C_k$ as I have in my answer. A colored permutation must have the color constant on its cycles so we get the number of possible colorings as $4^k$ with $k$ the number of cycles and from this point on you could continue as in my proof. – Marko Riedel Apr 05 '16 at 23:17
  • @MarkoRiedel: I have to say there aren't many differences between your solution and mine, beyond the phrasing. I still have the strong feeling that we are missing something really tricky that is able to show $L_{n+1}-L_n=(n+1)!-n!$ in a few lines. – Jack D'Aurizio Apr 05 '16 at 23:20
  • Maybe something like: if $\sigma\in S_n$ has order $2^m\cdot d$, by squaring $\sigma$ $m$ times we get an element with an odd order, that is associated with certain coloured permutations. But then? – Jack D'Aurizio Apr 05 '16 at 23:23
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    Addendum. What you have presented here is the combinatorial species $$\mathfrak{P}(\mathfrak{C}{\mathrm{odd}}(\mathcal{Z})) \mathfrak{P}(\mathfrak{C}{\mathrm{odd}}(\mathcal{Z})) \mathfrak{P}(\mathfrak{C}{\mathrm{odd}}(\mathcal{Z})) \mathfrak{P}(\mathfrak{C}{\mathrm{odd}}(\mathcal{Z}))$$ which is a perfectly valid approach to the problem and a genuine alternative to what I submitted. – Marko Riedel Apr 05 '16 at 23:57