arXiv subjects
Publications and source records attributed to Igor Pak.
We study partitions which attain the largest Littlewood-Richardson coefficient. More precisely, we prove that the largest $c^\lambda_{\mu\nu}$ is attained at partitions such that $\mu \subseteq \nu$ and $\nu/\mu$ is a disjoint union of squares. We conjecture that for $n \ge 16$, all largest $c^\lambda_{\mu\nu}$ must satisfy this property. We confirm this conjecture numerically, for $16 \le n \le 45$.
We give an algebraic branching program of polynomial size which computes Cayley determinant of right quantum matrices. This is a rare example of an efficient computation of a noncommutative determinant, and the first such example for quantum groups. We extend the results to the $q$-Cayley determinant of $q$-right quantum matrices, as well as to their multiparameter generalization. The proofs are entirely combinatorial, as we relate Cayley, Moore and Valiant determinants using bijections/involutions on words. We then employ the celebrated determinant construction of Mahajan and Vinay (SODA'97), to obtain the results.
We prove equality conditions for the Ahlswede--Daykin (AD) inequality and the Fortuin--Kasteleyn--Ginibre (FKG) inequality. We then present a number of applications and special cases of these equality conditions. These include Bj\"orner's and Fishburn's inequalities for linear extensions of finite posets, the Lam--Postnikov--Pylyavskyy (LPP) and the Okounkov inequalities for Schur positivity of products of Schur functions. We conclude with equality conditions for the Ahlswede--Daykin--Schur (ADS) inequality recently introduced in Chan--Chen--Pak--Soskin (2026), which is an AD type extension of the LPP inequality.
We generalize the Ahlswede--Daykin inequality (1978) to a Schur positive \emph{ADS inequality}, which also contains the Lam--Postnikov--Pylyavskyy inequality (2007) as a special case. We then present a number of further generalizations and applications. Notably, we resolve Mihalcea's conjecture on log-supermodularity of stable Grothendieck polynomials.
For a permutation $u\in S_n$, let $N\ast u\in S_{Nn}$ be the permutation with scaled Lehmer code. For given $u,v,w\in S_n$ and integer $N$, the stretched Schubert coefficients are defined as $f_{u,v,w}(N):=c_{N*u,N*v}^{N*w}$. Our main result is that the function $f_{u,v,w}(N)$ is eventually quasi-polynomial. This proves Kirillov's conjecture (2004), that the generating function for the sequence $\{f_{u,v,w}(N)\}$ is rational. For the proof, we use combinatorics of pipe dreams to show that Schubert coefficients are given as an alternating sum of the numbers of integer points in certain polytopes. These polytopes behave nicely under stretching, and we use Ehrhart theory to obtain the result. As a consequence of the proof, we also present new counterexamples to the saturation conjecture for Schubert coefficients, and give computational applications.
The saturation property for Littlewood--Richardson coefficients was established by Knutson and Tao in 1999. In 2004, Kirillov conjectured that the saturation property extends to Schubert coefficients. We disprove this conjecture in a strong form, by showing that it fails for a large family of instances. We also discuss computational complexity implications.
We give explicit positive combinatorial interpretations for the plethysm coefficients $\langle s_\mu[s_\nu], s_\lambda\rangle$, when $\lambda$ has at most two rows, as counting certain marked trees. In the special case $\mu=(n)$, this also yields a combinatorial interpretation for the corresponding rectangular Kronecker coefficient $g(\lambda, (n^k), (n^k))$. While it is easy to express these quantities as differences of counting problems in the complexity class $\mathrm{FP}$, putting the problem in $\#\mathrm{P}$, our interpretations give a positive counting formula over explicit marked trees.
The Schubert vanishing problem asks whether Schubert structure constants are zero. We give a complete solution of the problem from an algorithmic point of view, by showing that Schubert vanishing can be decided in probabilistic polynomial time.
We show that the Hilbert-Kunz multiplicity of the d-dimensional non-degenerate quadric hypersurface of characteristic p > 2 is a rational function of p composed from the Ehrhart polynomials of integer polytopes. In consequence, we prove that the Hilbert-Kunz multiplicity of quadrics of fixed characteristic is a decreasing function of dimension and recover results of Trivedi and Gessel-Monsky on the behaviour of said Hilbert-Kunz multiplicity as a function of characteristic.
We consider the decision problem of whether a particular Gromov--Witten invariant on a partial flag variety is zero. We prove that for the $3$-pointed, genus zero invariants, this problem is in the complexity class ${\sf AM}$ assuming the Generalized Riemann Hypothesis (GRH), and therefore lies in the second level of polynomial hierarchy ${\sf PH}$. For the proof, we construct an explicit system of polynomial equations through a translation of the defining equations. We also need to prove an extension of the Parametric Hilbert's Nullstellensatz to obtain our central reduction.
For a simple graph $G=(V,E)$ and edge $e\in E$, the effective resistance is defined as a ratio $\frac{\tau(G/e)}{\tau(G)}$, where $\tau(G)$ denotes the number of spanning trees in $G$. We resolve the inverse problem for the effective resistance for planar graphs. Namely, we determine (up to a constant) the smallest size of a simple planar graph with a given effective resistance. The results are motivated and closely related to our previous work arXiv:2411.18782 on Sedl\'a\v{c}ek's inverse problem for the number of spanning trees.
We give a signed puzzle rule to compute Schubert coefficients. The rule is based on a careful analysis of Knutson's recurrence arXiv:math/0306304. We use the rule to prove polynomiality of the sums of Schubert coefficients with bounded number of inversions.
The Schubert vanishing problem is a central decision problem in algebraic combinatorics and Schubert calculus, with applications to representation theory and enumerative algebraic geometry. The problem has been studied for over 50 years in different settings, with much progress given in the last two decades. We prove that the Schubert vanishing problem is in ${\sf AM}$ assuming the Generalized Riemann Hypothesis (GRH). This complements our earlier result in arXiv:2412.02064, that the problem is in ${\sf coAM}$ assuming the GRH. In particular, this implies that the Schubert vanishing problem is unlikely to be ${\sf coNP}$-hard, as we previously conjectured in arXiv:2412.02064. The proof is of independent interest as we formalize and expand the notion of a lifted formulation partly inspired by algebraic computations of Schubert problems, and extended formulations of linear programs. We use a result by Mahajan--Vinay to show that the determinant has a lifted formulation of polynomial size. We combine this with Purbhoo's algebraic criterion to derive the result.
Schubert coefficients $c_{u,v}^w$ are structure constants describing multiplication of Schubert polynomials. Deciding positivity of Schubert coefficients is a major open problem in Algebraic Combinatorics. We prove a positive rule for this problem based on two standard assumptions.
Schubert coefficients are nonnegative integers $c^w_{u,v}$ that arise in Algebraic Geometry and play a central role in Algebraic Combinatorics. It is a major open problem whether they have a combinatorial interpretation, i.e, whether $c^w_{u,v} \in \#{\sf P}$. We study the closely related vanishing problem of Schubert coefficients: $\{c^w_{u,v}=^? 0\}$. Until this work it was open whether this problem is in the polynomial hierarchy ${\sf PH}$. We prove that $\{c^w_{u,v}=^? 0\}$ in ${\sf coAM}$ assuming the GRH. In particular, the vanishing problem is in ${\Sigma_2^{{\text{p}}}}$. Our approach is based on constructions lifted formulations, which give polynomial systems of equations for the problem. The result follows from a reduction to Parametric Hilbert's Nullstellensatz, recently studied in arXiv:2408.13027. We extend our results to all classical types. Type $D$ is resolved in the appendix (joint with David Speyer).
We prove the exponential growth of the cardinality of the set of numbers of spanning trees in simple (and planar) graphs on $n$ vertices, answering a question of Sedl\'a\v{c}ek from 1969. The proof uses a connection with continued fractions, ``thin orbits,'' and Zaremba's conjecture.
We give an explicit counterexample to the Bunkbed Conjecture introduced by Kasteleyn in 1985. The counterexample is given by a planar graph on $7222$ vertices, and is built on the recent work of Hollom (2024).
We consider a probabilistic version of the depth-first search on mazes with two exits, and show that this algorithm has equal probability of finding either exit. The proof is combinatorial and uses an explicit involution.