# Generating functions for Hankel determinants of Catalan numbers

The Hankel determinants of the Catalan numbers are well known and can be written as $$d(k,n)= \det \left( C_{k + i + j} \right)_{i,j = 0}^{n - 1}=\prod_{i=1}^{k-1}\frac{\binom{2n+2j}{j}}{\binom{2j}{j}}$$ with $$d(k,0)=1.$$

Computations suggest that $$D_k(x)=\sum_{n\geq 0}d(k,n)x^n=\frac{A_{k}(x)}{(1-x)^{\binom{k}{2}+1}}$$ where $$A_{k}(x)$$ is a palindromic and unimodal polynomial of degree $$\binom{k-1}{2}.$$

Moreover it seems that $$A_{k}(x)$$ is gamma-nonnegative, i.e. a linear combination of polynomials $$x^j(1+x)^{\binom{k-1}{2}-2j}$$ with positive coefficients.

Is this known? Any idea how to prove this?

At any rate, this interpretation means that your generating function $$D_k(x)$$ is the same as the generating function $$\sum_{m \geq 0}\Omega_P(m)x^m$$ where $$\Omega_P(m)$$ counts the number of order preserving maps $$P\to \{0,1,\ldots,m\}$$ for a certain poset $$P$$ (namely, the staircase partition shape poset; equivalently, the Type A root poset). The general theory of $$P$$-partitions, as developed by Stanley, thus says that $$D_k(x) = \sum_{L} x^{\mathrm{des}(L)}/(1-x)^{\binom{k}{2}-1}$$, where the sum is over linear extensions of $$P$$, when it is naturally labeled (see e.g. Theorem 3.15.8 of EC1). Thus your $$A_k(x)$$ is the $$P$$-version of an Eulerian number generating function, and a general result of Brändén (see https://doi.org/10.37236/1866 or https://arxiv.org/abs/1410.6601) says that such poset linear extension descent genearting functions are $$\gamma$$-nonnegative as long as the poset in question is graded, which it is in this case.
• @Sam Hopkins: Thank you for this information. There is an analogous situation for the Hankel determinants $c(k,n)= \det \left( b_{k + i + j} \right)_{i,j = 0}^{n - 1}$ of $b(n)= \binom{n}{\lfloor{\frac{n}{2}}\rfloor}.$ Let $C_k(x)=\sum_{n\geq 0}c(2k,n)x^n=\frac{B_{k}(x)}{(1-x)^{k^2+1}}$ be the generating function of $c(2k,n)=\prod_{i=1}^n\prod_{j=0}^{k-1}\frac{k+i+j}{i+j}.$ Then it seems that $B_k(x)$ is also gamma positive. Is there an analogous interpretation? Aug 30, 2021 at 6:45
• @JohannCigler: Yes, there is an analogous interpretation. This time $P$ will be the Type B root poset (alias "shifted double staircase shape"). The fact that this second Hankel determinant counts such bounded $P$-partitions is closely related to the fact that so-called "Dyck path prefixes" are counted by $\binom{n}{\lfloor n/2 \rfloor}$ (see e.g. arxiv.org/abs/1406.1709); briefly, we now consider fans of Dyck path prefixes. Incidentally, these kind of bounded $P$-partitions were again first counted by Proctor, in doi.org/10.2307/2045516. Aug 30, 2021 at 11:40