Parking cars of different sizes, The American Mathematical Monthly, Vol. 123, No. 10 (December 2016), pp. 1045–1048 (with Richard Ehrenborg), 2016.
We extend the notion of parking functions to parking sequences, which include cars of different sizes, and prove a product formula for the number of such sequences.
Parking cars after a trailer, Australasian Journal of Combinatorics, Vol. 70, No. 3 (February 2018), pp. 402–406 (with Richard Ehrenborg), 2018.
Using the notion of parking sequences given previously, we here give a refinement of that result involving parking the cars after a trailer. The proof of the refinement uses a multi-parameter extension of the Abel-Rothe polynomial due to Strehl.
On the powers of the descent set statistic, Advances in Applied Mathematics, Vol. 96 (May 2018), pp. 1–17 (with Richard Ehrenborg), 2018.
We study the sum of the rth powers of the descent set statistic and how many small prime factors occur in these numbers. Our results depend upon the base p expansion of n and r.
Box polynomials and the excedance matrix, submitted to Discrete Mathematics (with Richard Ehrenborg, Dustin Hedmark, and Cyrus Hettle), 24 pages.
We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions λ whose Young diagrams fit in an m by n box. We show that these polynomials can be expressed by the finite difference operator applied to the power xm+n. Evaluating box polynomials yields a variety of identities involving set partition enumeration. We extend the latter identities using restricted growth words and a new operator called the fast Fourier operator, and consider connections between set partition enumeration and the chromatic polynomial on graphs. We also give connections between the box polynomials and the excedance matrix, which encodes combinatorial data from a noncommutative quotient algebra motivated by the recurrence for the excedance set statistic on permutations.
The boustrophedon transform for descent polytopes, submitted to European Journal of Combinatorics (with Richard Ehrenborg), 4 pages.
We give a short proof that the f‐vector of the descent polytope DPv is componentwise maximized when the word v is alternating. Our proof uses an f‐vector analogue of the boustrophedon transform.
The antipode of the noncrossing partition lattice, submitted to Advances in Applied Mathematics (with Richard Ehrenborg), 10 pages.
We prove the cancellation-free formula for the antipode of the noncrossing partition lattice in the reduced incidence Hopf algebra of posets due to Einziger. The proof is based on a map from chains in the noncrossing partition lattice to noncrossing hypertrees and expressing the alternating sum over these fibers as an Euler characteristic.