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Let's consider the sequence of coefficients of $\prod_{i}\frac {1-x^{d_i}} {1-x}$, where $d_i$ is a monotonically increasing nonnegative integer sequence. How to prove that the coefficients form an arithmetic triangle $T(n,k)$ and how to show the unimodality of its $n$th row?

E.g. if $d_i$ are natural numbers, then the product of polynomials has the property of unimodality (for the case $T(n,k)$ is the number of permutations of $\{1\dotsc n\}$ with $k$ inversions). Moreover, there are some examples for groups $A_n$, $B_n$, $D_n$.

I tried other sequences like Fibonacci numbers $F(n)$, Prime numbers $p_n$, triangular numbers and other famous integer sequences. Each time I get the arithmetic triangle. Any ideas how to prove the property in general?

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    $\begingroup$ I don't know what "arithmetic triangle" means. But the product of two polynomials with nonnegative, symmetric, unimodal coefficients is well known to have nonnegative, symmetric, unimodal coefficients so that immediately explains the unimodality you observe. $\endgroup$
    – Sam Hopkins
    Commented Feb 6, 2023 at 19:14
  • $\begingroup$ All, the factors are so-called q-integers, which are symmetric and unimodal as Sam says. $\endgroup$
    – Per Alexandersson
    Commented Feb 6, 2023 at 19:29

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