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Some friends and I have a family of polynomials (in one variable) with rational coefficients and we would very much like a formula for them. Grasping at straws, we computed many examples and wrote them in the basis of binomial coefficients. Specifically, I mean the basis $\left\{\binom{x}{0},\binom{x}{1},\binom{x}{2},\ldots\right\}$ of the ring of rational polynomials over $x$. We were surprised to find that our polynomials all expand positively and integrally in that basis.

That is, if one of our polynomials $p(x)$ of degree d is written as $\sum_{k=0}^da_k\binom{x}k$, each of the $a_k$ is a nonnegative integer.

We can't make much sense of the coefficients. But we're wondering if the positivity of our polynomials in the binomial coefficients basis is a sort of "shadow" of some other stronger phenomenon. Suppose there were some other basis $\{b_0(x),b_1(x),\ldots\}$ such that each $b_i(x)$ expands nonnegatively in the basis of binomial coefficients. If our polynomials expand nonnegatively (and in some understandable way) in the basis $\{b_0(x),b_1(x),\ldots\}$, that's our desired formula.

If you're thinking "They're still grasping at straws", you're right. But it can't hurt to ask:

What bases for the polynomial ring should we try? Are there some well-known bases that expand positively in the binomial coefficients basis?

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    $\begingroup$ Not directly answering your question, but if you have a polynomial that expands nonnegatively into binomials, it is often the Hilbert polynomial of an interesting Cohen-Macaulay graded algebra. $\endgroup$
    – Sam Hopkins
    Commented Sep 30, 2021 at 20:39
  • $\begingroup$ Have you checked for unimodality/log-concavity? If your polynomials are of combinatorial nature you could try expressing them as chromatic polynomials of (insert adjective) graphs. $\endgroup$
    – Gjergji Zaimi
    Commented Sep 30, 2021 at 20:44
  • $\begingroup$ Sorry, I have been looking and these with cleared denominators too long, so I asked the question slightly wrong at first. The polynomials have rational coefficients, but they expand as nonnegative integer combinations of binomial coefficients. I edited the question. I don't think this fundamentally alters what I'm asking, but sorry to get it wrong! $\endgroup$
    – Nathan Reading
    Commented Oct 1, 2021 at 0:24
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    $\begingroup$ Maybe at least you could show the first few, of lowest degrees, say, up to 10? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Oct 1, 2021 at 11:59
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    $\begingroup$ @მამუკაჯიბლაძე: My question is "What bases for the polynomial ring should we try? Are there some well-known bases that expand positively in the binomial coefficients basis?" I think that's a reasonable question about research methods, and it may be of general interest. Too many specifics of the polynomials would muddy the waters regarding this question about methods. Also, I am not asking the question "Can anybody figure out these polynomials for us?", and I would rather not implicitly ask the mathoverflow community to analyze the polynomials. $\endgroup$
    – Nathan Reading
    Commented Oct 1, 2021 at 18:06

2 Answers 2

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The basis $\binom{x}{0}$, $\binom{x+1}{1}$, $\binom{x+2}{2}$, $\dots$ has this property. More generally, if $i$ is a nonnegative integer then $\binom{x+i}{j}$ is a nonnegative linear combination of the $\binom{x}{k}$.

A useful basis for polynomials of degree at most $n$ with this property is $\binom{x}{n}$, $\binom{x+1}{n}$, $\dots$, $\binom{x+n}{n}$.

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  • $\begingroup$ Thanks, we'll look at these possibilities. $\endgroup$
    – Nathan Reading
    Commented Oct 1, 2021 at 0:37
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We conjecture that the coefficients of Jack polynomials can be expressed nicely in this basis, see https://arxiv.org/pdf/1810.12763.pdf

Also, there is a close connection with rook polynomials and hit polynomials, as well as the relation between Ehrhart polynomials and the $h^*$-vector, which uses this type of polynomials.

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