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Let $n$ and $k$ be natural numbers. I will consider North-East lattice paths (NE-paths) from $(0,0)$ to $(n,n)$ and encode these as strings of length $2n$ with letters $\mathsf{N}$ and $\mathsf{E}$. A peak of such a lattice path $\mathsf{P} = \mathsf{x_1x_2\dots x_{2n}}$ is an index $i$ such that $\mathsf{x_i x_{i+1}}=\mathsf{NE}$. The major index of $P$ is the sum of the peaks of $\mathsf{P}$. As an example, the NE-path $\mathsf{P} = \mathsf{ENNENEEENENN}$ has peaks at indices $3,5$ and $9$ and hence $\text{maj}(\mathsf{P})=3+5+9=17$. I have two questions regarding these paths.

  1. From this OEIS-entry, I understand that the number NE-paths from $(0,0)$ to $(n,n)$ with exactly $k$ peaks is equal to $\binom{n}{k}^2$. Does someone know a proof or a reference for this?
  2. I suspect that the $\text{maj}$-generating polynomial over this set of paths has the following nice closed form expression: $$\sum_\mathsf{P}q^{\text{maj}(\mathsf{P})}=q^{k^2}\begin{bmatrix} n\\ k \end{bmatrix}_q^2.$$ Here, the sum is taken over all NE-lattice paths from $(0,0)$ to $(n,n)$ with exactly $k$ peaks. I feel like this is a nice enough formula that it should be mentioned somewhere in the litterature but I have not managed to find it anywhere. Does someone know a reference for this?
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    $\begingroup$ 1. This is equivalent to Exercise 3 (a) in UMN Fall 2018 Math 5705 midterm #3, since you can encode NE-paths from $\left(0,0\right)$ to $\left(n,n\right)$ as $2n$-tuples with entries in $\left\{0,1\right\}$ having exactly $n$ $1$-positions (just replacing $N$ and $E$ by $1$ and $0$, respectively). $\endgroup$ – darij grinberg Nov 15 '19 at 23:11
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    $\begingroup$ For 1, the paths with $k$ peaks are in an one-to-one correspondence with the pairs of subsets $X, Y\subset [n]$ with $|X|=|Y|=k$. To see this, just let $X$ be the set of $x$-coordinates of the peaks in a path, $Y$ be the set of their $y$-coordinates. So, the total number of paths with $k$ peaks is $\binom{n}{k}^2$. $\endgroup$ – Max Alekseyev Nov 15 '19 at 23:30
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The formula in 2 is a very special case of a result of Richard Stanley's, though it certainly could be older. (I wouldn't be surprised if it can be found in MacMahon's work.) See, e.g., my paper A historical survey of P-partitions, section 7.2, for references. Just in case the connection isn't clear, your problem is equivalent to counting shuffles of the words $00\cdots 0$ and $11\cdots1$, both of length $n$, with $k$ descents, by major index. Stanley's formula counts shuffles of two arbitrary (but disjoint) permutations with a given number of descents by major index.

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Imagine that you lay out the N (0) and E (1) moves as follows ($n=4$ shown): $$0000$$ $$1111$$ As you go along the path, color $\color{red}{red}$ the ones you have used, so that after reading either 001, 010, or 100 you have: $${\color{red}{00}}00$$ $${\color{red}1}111$$ Then you have to choose after which 0s to switch to 1s, and which 1s to switch to 0s, for a total of $\binom{n}{k}^2$ ways.

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