A finite sequence $a_i$ is called logconvace in case $a_i^2 \geq a_{i-1} a_{i+1}$.
Question : For a fixed $n$, is the sequence $a_{n,k}$ giving the number of Dyck paths of semilength $n$ having height $k$ logconcave? (see http://oeis.org/A080936)
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4$\begingroup$ A starting point is the product formula $(2k^2+6k+1-3n)(2n)!/((n-k)!(n+k+3)!)$ given in the reference, which is, however, valid only for $(n+1)/2\leq k\leq n$. Computing $T(n,k-1)T(n,k+1)/T(n,k)^2$ gives you three natural factors, which are all less than $1$. $\endgroup$– Martin RubeyCommented Mar 26, 2020 at 20:55
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$\begingroup$ I have an idea for a combinatorial approach that I'll just sketch for now. (1) Given two height $k$ paths, follow the first path until you reach height $k$, then insert the entire second path, and finally complete the first path. (2) Given a height $k-1$ path and a height $k+1$ path, build the same kind of composite path by interrupting the first at height $k-1$. Both of these approaches create height $2k$ paths. Why does (1) generate more? Or starting from the height $2k$ paths, why are there more ways to split them into two height $k$ paths than heights $k-1$ and $k+1$ paths? $\endgroup$– Brian HopkinsCommented Jun 13, 2020 at 20:21
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1 Answer
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A stronger property than log-concavity, is real-rootedness of $\sum_k t^k a_{n,k}$. However, for $n=4$, this polynomial is $1 + 7 t + 5 t^2 + t^3$ which is not real-rooted.
answered Jun 8, 2020 at 19:11