16 citations · 69 across the 23 of their papers we have counts for
24 papers
Graph-like Scheduling Problems and Property B
John Machacek
Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like schedu…
-Rational and -Real Binomial Coefficients
John Machacek, Nicholas Ovenhouse
We consider -binomial coefficients built from the -rational and -real numbers defined by Morier-Genoud and Ovsienko in terms of continued fractions. We establish versions…
Red sizes of quivers
Eric Bucher, John Machacek
In this article, we will expand on the notions of maximal green and reddening sequences for quivers associated to cluster algebras. The existence of these sequences has been studie…
Mechanical proving with Walnut for squares and cubes in partial words
John Machacek
Walnut is a software that using automata can prove theorems in combinatorics on words about automatic sequences. We are able to apply this software to both prove new results as wel…
Discrete Dynamical Systems From Real Valued Mutation
John Machacek, Nicholas Ovenhouse
We introduce a family of discrete dynamical systems which includes, and generalizes, the mutation dynamics of rank two cluster algebras. These systems exhibit behavior associated w…
Lattice walks ending on a coordinate hyperplane avoiding backtracking and repeats
John Machacek
We work with lattice walks in using step set that finish with . We further impose conditions of avoiding backtracking (i.e. $[v,-v…
Shelling the m=1 amplituhedron
Steven N. Karp, John Machacek
The amplituhedron was introduced by Arkani-Hamed and Trnka (2014) in order to give a geometric basis for calculating scattering amplitudes in planar $\mathcal…
Unique maximum independent sets in graphs on monomials of a fixed degree
John Machacek
We consider graphs on monomials in variables of a fixed degree where two monomials are adjacent if and only if their least common multiple has degree . We prove that w…
Sign variation and descents
Nantel Bergeron, Aram Dermenjian, John Machacek
For any and , let be the poset of projective equivalence classes of -vectors of length with sign variation bounded by , ordered by…
Boundary measurement and sign variation in real projective space
John Machacek
We define two generalizations of the totally nonnegative Grassmannian and determine their topology in the case of real projective space. We find the spaces to be PL manifolds with…
New Invariants for Permutations, Orders and Graphs
Jean-christophe Aval, Nantel Bergeron, John Machacek
We study the symmetric function and polynomial combinatorial invariants of Hopf algebras of permutations, posets and graphs. We investigate their properties and the relations among…
Building maximal green sequences via component preserving mutations
Eric Bucher, John Machacek, Evan Runburg +2
We introduce a new method for producing both maximal green and reddening sequences of quivers. The method, called component preserving mutations, generalizes the notion of direct s…
Combinatorial interpretations of Lucas analogues of binomial coefficients and Catalan numbers
Curtis Bennett, Juan Carrillo, John Machacek +1
The Lucas sequence is a sequence of polynomials in s, and t defined recursively by {0}=0, {1}=1, and {n}=s{n-1}+t{n-2} for n >= 2. On specialization of s and t one can recover the…
Partial words with a unique position starting a square
John Machacek
We consider partial words with a unique position starting a power. We show that over a letter alphabet, a partial word with a unique position starting a square can contain at m…
Strong factorization and the braid arrangement fan
John Machacek
We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. Our method uses a correspondence between cones and preposets.
Reddening Sequences for Banff Quivers and the Class
Eric Bucher, John Machacek
We show that a reddening sequence exists for any quiver which is Banff. Our proof is combinatorial and relies on the triangular extension construction for quivers. The other facts…
Upper cluster algebras and choice of ground ring
Eric Bucher, John Machacek, Michael Shapiro
We initiate a study of the dependence on the choice of ground ring on the question of whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is…
Hypergraphic polytopes: combinatorial properties and antipode
Carolina Benedetti, Nantel Bergeron, John Machacek
In an earlier paper, the first two authors defined orientations on hypergraphs. Using this definition we provide an explicit bijection between acyclic orientations in hypergraphs a…
Egyptian Fractions and Prime Power Divisors
John Machacek
From varying Egyptian fraction equations we obtain generalizations of primary pseudoperfect numbers and Giuga numbers which we call prime power psuedoperfect numbers and prime powe…
Log-Canonical Coordinates for Poisson Brackets and Rational Changes of Coordinates
John Machacek, Nicholas Ovenhouse
Goodearl and Launois have shown that for a log-canonical Poisson bracket on affine space there is no rational change of coordinates for which the Poisson bracket is constant. Our m…
Plurigraph coloring and scheduling problems
John Machacek
We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acycli…
Boundary Measurement Matrices for Directed Networks on Surfaces
John Machacek
Franco, Galloni, Penante, and Wen have proposed a boundary measurement map for a graph on any closed orientable surface with boundary. We consider this boundary measurement map whi…
Deleting Powers in Words
John Machacek
We consider the language consisting of all words such that it is possible to obtain the empty word by iteratively deleting powers. It turns out that in the case of deleting squares…
Combinatorial Hopf Algebras of Simplicial Complexes
Carolina Benedetti, Joshua Hallam, John Machacek
We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a fa…