Based on the Article: Cyclotomic Generating Functions, written by Sara Billey and Joshua P. Swanson. Revision as of May 10, 2023.
Abstract: It is a remarkable fact that for many statistics on finite sets of combinatorial objects, the roots of the corresponding generating function are each either a complex root of unity or zero. We call such polynomials cyclotomic generating functions (CGF's). In prior work, the authors studied the support and asymptotic distribution of the coefficients of several families of CGF's arising from tableau and forest combinatorics. In this paper, we continue these explorations by studying general CGF's from algebraic, analytic, and asymptotic perspectives. We review some of the many known examples of CGF's; describe their coefficients, moments, cumulants, and characteristic functions; and give a variety of necessary and sufficient conditions for their existence arising from probability, commutative algebra, and invariant theory. We further show that CGF's are ``generically'' asymptotically normal, generalizing a result of Diaconis. We include several open problems concerning CGF's. Code for the data included in the Appendix is available below.
Companion Code: CGFs.ipynb. This notebook should be run on a SageMath kernel. It has been tested on SageMath 9.8.
This notebook includes routines for testing if a univariate polynomial with nonnegative integer coefficients is a CGF, for converting CGF's to cyclotomic or rational forms, and for iterating over the elements and minimal generating sets of the cyclotomic monoids found in the paper, including the monoid of all basic cyclotomic polynomials, unimodal CGFs, log-concave CGFs, the Gale monoid, and the monoid corresponding to homogenious systems of parameters. In particular, this code may be used to regenerate the tables in the Appendix and the associated OEIS entries:| M | |M_n| | Irreducibles |
|---|---|---|
| Log-concave monoid | A360622 | A361439 |
| Unimodal monoid | A360621 | A361440 |
| Gale monoid | A362553 | A362554 |
| Basic CGF monoid | A360620 | A361441 |
| Cyclotomic monoid | A120963 | A014197 |