arXiv subjects
Publications and source records attributed to Molly W. Dunkum.
The move-minimizing puzzles presented here are certain types of one-player combinatorial games that are shown to have explicit solutions whenever they can be encoded in a certain way as diamond-colored modular or distributive lattices. Our work here is founded in a new interpretation of some routine and elementary order-theoretic combinatorics.
We construct every finite-dimensional irreducible representation of the simple Lie algebra of type $\mathsf{E}_{7}$ whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type $\mathsf{E}_{7}$ root system. As a consequence, we obtain constructions of each finite-dimensional irreducible representation of the simple Lie algebra of type $\mathsf{E}_{6}$ whose highest weight is a nonnegative integer linear combination of the two dominant minuscule $\mathsf{E}$-weights. Our constructions are explicit in the sense that, if the representing space is $d$-dimensional, then a weight basis is provided such that all entries of the $d \times d$ representing matrices of the Chevalley generators are obtained via explicit, non-recursive formulas. To effect this work, we introduce what we call $\mathsf{E}_{6}$- and $\mathsf{E}_{7}$-polyminuscule lattices that analogize certain lattices associated with the famous special linear Lie algebra representation constructions obtained by Gelfand and Tsetlin.
`Terquem's problem' is a name given in the twentieth century to the problem of enumerating certain integer sequences whose entries alternate in parity. In particular, this problem asks for the count of strictly increasing length $m$ sequences of positive integers bounded above by some integer $n$ whose odd-indexed entries are odd and whose even-indexed entries are even. This problem and its generalizations have been well-studied. However, the putative original source for this problem, an 1839 paper by Olry Terquem, is subtly different from the problem that is now attributed to Terquem. In this paper, we highlight this distinction and also make connections between Terquem's `forgotten' problem and the Fibonacci sequence, continuants, Bézout's Lemma, and the extended version of the Euclidean Algorithm.
We present a family of rank symmetric diamond-colored distributive lattices that are naturally related to the Fibonacci sequence and certain of its generalizations. These lattices re-interpret and unify descriptions of some un- or differently-colored lattices found variously in the literature. We demonstrate that our symmetric Fibonaccian lattices naturally realize certain (often reducible) representations of the special linear Lie algebras, with weight basis vectors realized as lattice elements and Lie algebra generators acting along the covering digraph edges of each lattice. We present evidence that each such weight basis possesses certain distinctive extremal properties. We provide new descriptions of the lattice cardinalities and rank generating functions and offer several conjectures/open problems. Throughout, we make connections with integer sequences from the OEIS.
We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of $\mathfrak{sl}(n,\mathbb{C})$ obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for each finite-dimensional representation of $\mathfrak{sl}(n,\mathbb{C})$ associated to a given skew Schur function. Our constructions use diamond-colored distributive lattices of skew-shaped semistandard tableaux that generalize some classical Gelfand--Tsetlin (GT) lattices. Our constructions take place within the context of a certain programmatic study of poset models for semisimple Lie algebra representations and Weyl group symmetric functions undertaken by the first-named author and others. Some key aspects of the methodology of that program are recapitulated here. Combinatorial and representation-theoretic applications of our constructions are pursued here and elsewhere.
We consider various properties and manifestations of some sign-alternating univariate polynomials borne of right-triangular integer arrays related to certain generalizations of the Fibonacci sequence. Using a theory of the root geometry of polynomial sequences developed by J. L. Gross, T. Mansour, T. W. Tucker, and D. G. L. Wang, we show that the roots of these `sign-alternating Gibonacci polynomials' are real and distinct, and we obtain explicit bounds on these roots. We also derive Binet-type closed expressions for the polynomials. Some of these results are applied to resolve finiteness questions pertaining to a one-player combinatorial game (or puzzle) modelled after a well-known puzzle we call the `Networked-numbers Game.' Elsewhere, the first- and second-named authors, in collaboration with A. Nance, have found rank symmetric `diamond-colored' distributive lattices naturally related to certain representations of the special linear Lie algebras. Those lattice cardinalities can be computed using sign-alternating Fibonacci polynomials, and the lattice rank generating functions correspond to the rows of some new and easily defined triangular integer arrays. Here, we present Gibonaccian, and in particular Lucasian, versions of those symmetric Fibonaccian lattices/results, but without the algebraic context of the latter.
A "truncation" of Pascal's triangle is a triangular array of numbers that satisfies the usual Pascal recurrence but with a boundary condition that declares some terminal set of numbers along each row of the array to be zero. Presented here is a family of natural truncations of Pascal's triangle that generalize a kind of Catalan triangle. The numbers in each array are realized as differences of binomial coefficients, as counts of certain lattice paths and tableaux, and as entries of representing matrices for certain linear transformations of polynomial spaces. Lucas's theorem is applied to determine precisely those truncations for which the number of odd entries on each row is a power of two.