The $d$-sequence conjecture for coefficients of uniform-matroid Kazhdan–Lusztig polynomials

For integers $m$ and $d$, and $0\leq i\leq d$, define

$$f_m(d,i)=\sum_{h=0}^{m-1}\frac{1}{(m-h)\binom{m}{h}}\binom{i+m}{m-h-1}\binom{i-1+h}{h}\binom{d-i+h}{h}.$$

A finite sequence of real numbers is an $n$-sequence when the associated polynomial has real zeros of one common sign. The $d$-sequence conjecture. The sequence $\{f_m(d,i)\}_{i=0}^{d}$ is a $d$-sequence for every $m$ and $d$. Numerical evidence supports the conjecture, and the paper verifies the needed cases $2\leq m\leq 15$ to establish real-rootedness there; the general assertion remains open.