Submitted by Sam Hopkins Let be a poset. The Dedekind-MacNeille completion of is the “smallest” complete lattice with an embedding , in the sense that is a subposet of any other such lattice. (We will exclusively be interested in finite posets and lattices, for which completeness is automatic, so from now on we will drop the adjective complete.) It can be constructed explicitly from in much the same way that the real numbers can be obtained from the rationals using Dedekind cuts, hence the name. Consider the case , the (strong) Bruhat order on the symmetric group. Unlike weak order, Bruha ..read more

The University of Minnesota’s Open Problems in Algebraic Combinatorics conference (which was originally planned for 2020 but was pushed back two years because of the COVID-19 pandemic) will take place in person May 16-20 of 2022. See the conference website for more information. Note that the conference will be accompanied by a proceedings to be published by the AMS, and some of the articles for the proceedings are now available on the conference website. These articles also discuss open problems in algebraic combinatorics, in somewhat more detail than the posts featured on this blog ..read more

Submitted by Eric Bucher and John Machacek Cluster algebras are commutative rings defined by Fomin and Zelevinsky [1] with a distinguished set of generators determined by a combinatorial process known as mutation. These algebras are known to be connected with various areas of mathematics and physics. Hence, the theory of cluster algebras has many directions and open problems. Here, we will focus on some purely combinatorial open problems in cluster algebra theory with the hope of giving researchers in combinatorics an avenue into cluster algebras. Also, we think these problems will interest cl ..read more

Submitted by Gleb Nenashev Zonotopal algebras were defined for hyperplane arrangements independently by F. Ardila and A. Postnikov in [2] and O. Holtz and A. Ron in [3]. Prior to these works A. Postnikov and B. Shapiro defined the family of graphical algebras; see [5, 6]. Here we present definitions and the problem only for the graphical case. For the case of unimodular zonotopal algebras and general zonotopal algebras, all definitions, results, and the conjecture can be extended; see more details in [4]. Let be a graph on vertices. We index its vertices with numbers to , i.e., . For an int ..read more

Submitted by Mike Zabrocki Parts of this open problem are well studied and the results are well known, other aspects and variations have been barely explored. Start with the polynomial ring in sets of commuting variables and sets of anti-commuting variables . That is there are variables and with , , satisfying the relations for all , and . That is we are looking at the ring of polynomials in these variables and we will denote this polynomial ring as . Now let be a group which acts on the first index of the variables. That is, for each , there exists coefficients and such that for ..read more

Submitted by Maria Monks Gillespie There are many natural combinatorial problems yet to be solved in the study of two-parameter symmetric functions such as Macdonald polynomials. We describe two of them here, both of which ask to explain the symmetry between and exhibited by certain combinatorially defined polynomials in and . For the first, a good general reference for the notions involved is [2], and for the second, [4]. Diagonal coinvariants and parking functions Consider the diagonal action of the symmetric group on , in which permutations act simultaneously on the two sets of variable ..read more

Submitted by Per Alexandersson and Valentin Féray Let be a positive real parameter and consider the Jack polynomials , indexed by partitions . Jack polynomials are standard deformations of Schur functions, which can be defined using either a scalar product or differential operators. They are a degenerate case of the celebrated Macdonald polynomials. For background, we refer to [4] and [10], from which we borrow our notation. A natural question is how these polynomials multiply, i.e. we want to investigate the coefficients defined by . To state our first open problem, we need to introduce two ..read more

Submitted by Dennis White Suppose is a finite permutation group acting on . Let denote the conjugacy class of permutations of type in symmetric group . Let be the irreducible character evaluated at the conjugacy class . Define In fact, is the number of occurrences of the irreducible in the induction of the trivial character of up to , or, by Frobenius reciprocity, the dimension of the -fixed space inside the -irreducible corresponding to . It is therefore an integer and where is the number of standard Young tableaux (SYT) of shape . OPAC-024. Interpret as a subset of SYT of shape ..read more

Submitted by Joel Brewster Lewis Here is one collection of permutation statistics associated to a permutation in the symmetric group , viewed as a sequence containing each element of exactly once: for any , let be the maximum size of the disjoint union of increasing subsequences of . For example, if , then , (witnessed uniquely by the subsequences ), (witnessed uniquely by the pair of subsequences ), and for all . Similarly, one can define a second collection of permutation statistics by instead taking decreasing subsequences; with one has , , , , and for all . The following paraphra ..read more