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Igor Pak

Publications and source records attributed to Igor Pak.

At least 91 records · Page 5Linked to original sources
arXiv2017

Cutoff for biased transpositions

In this paper we study the mixing time of a biased transpositions shuffle on a set of $N$ cards with $N/2$ cards of two types. For a parameter $0<a \le 1$, one type of card is chosen to transpose with a bias of $\frac{a}{N}$ and the other type is chosen with probability $\frac{2-a}{N}$. We show that there is cutoff for the mixing time of the chain at time $\frac{1}{2a} N \log N$. Our proof uses a modified marking scheme motivated by Matthews' proof of a strong uniform time for the unbiased shuffle.

math.PR↗
arXiv2017

The computational complexity of integer programming with alternations

We prove that integer programming with three quantifier alternations is $NP$-complete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two quantifier alternations can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes $P,Q \subset \mathbb{R}^4$ , counting the projection of integer points in $Q \backslash P$ is $\#P$-complete. This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of integer points in $P$ and $Q$ separately.

math.CO↗
arXiv2017

Complexity of short Presburger arithmetic

We study complexity of short sentences in Presburger arithmetic (Short-PA). Here by "short" we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integers involved in the inequalities. We prove that assuming Kannan's partition can be found in polynomial time, the satisfiability of Short-PA sentences can be decided in polynomial time. Furthermore, under the same assumption, we show that the numbers of satisfying assignments of short Presburger sentences can also be computed in polynomial time.

math.CO↗
arXiv2017

Asymptotics of the number of standard Young tableaux of skew shape

We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse's hook-length formula. We also compare our bounds with the existing bounds on the numbers of linear extensions of the corresponding posets.

math.CO↗
arXiv2016

Limit shapes via bijections

We compute the limit shape for several classes of restricted integer partitions, where the restrictions are placed on the part sizes rather than the multiplicities. Our approach utilizes certain classes of bijections which map limit shapes continuously in the plane. We start with bijections outlined previously by the second author, and extend them to include limit shapes with different scaling functions.

math.CO↗
arXiv2016

Fast domino tileability

Domino tileability is a classical problem in Discrete Geometry, famously solved by Thurston for simply connected regions in nearly linear time in the area. In this paper, we improve upon Thurston's height function approach to a nearly linear time in the perimeter.

math.CO↗
arXiv2016

Bounds on Kronecker and $q$-binomial coefficients

We present a lower bound on the Kronecker coefficients for tensor squares of the symmetric group via the characters of~$S_n$, which we apply to obtain various explicit estimates. Notably, we extend Sylvester's unimodality of $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in~$q$ to derive sharp bounds on the differences of their consecutive coefficients. We then derive effective asymptotic lower bounds for a wider class of Kronecker coefficients.

math.CO↗
arXiv2015

On Higman's $k(U_n(\mathbb{F}_q))$ conjecture

A classical conjecture by Graham Higman states that the number of conjugacy classes of $U_n(q)$, the group of upper triangular $n\times n$ matrices over $\mathbb{F}_q$, is polynomial in $q$, for all $n$. In this paper we present both positive and negative evidence, verifying the conjecture for $n\le 16$, and suggesting that it probably fails for $n\ge 59$. The tools are both theoretical and computational. We introduce a new framework for testing Higman's conjecture, which involves recurrence relations for the number of conjugacy classed of \emph{pattern groups}. These relations are proved by the \emph{orbit method} for finite nilpotent groups. Other applications are also discussed.

math.CO↗
arXiv2015

Pattern avoidance is not P-recursive

Let $F \subset S_k$ be a finite set of permutations and let $C_n(F)$ denote the number of permutations $σ$ in $S_n$ avoiding the set of patterns $F$. The Noonan-Zeilberger conjecture states that the sequence ${C_n(F)}$ is P-recursive. We use Computability Theory to disprove this conjecture.

math.CO↗
arXiv2015

On the complexity of computing Kronecker coefficients

We study the complexity of computing Kronecker coefficients $g(λ,μ,ν)$. We give explicit bounds in terms of the number of parts $\ell$ in the partitions, their largest part size $N$ and the smallest second part $M$ of the three partitions. When $M = O(1)$, i.e. one of the partitions is hook-like, the bounds are linear in $\log N$, but depend exponentially on $\ell$. Moreover, similar bounds hold even when $M=e^{O(\ell)}$. By a separate argument, we show that the positivity of Kronecker coefficients can be decided in $O(\log N)$ time for a bounded number $\ell$ of parts and without restriction on $M$. Related problems of computing Kronecker coefficients when one partition is a hook, and computing characters of $S_n$ are also considered.

math.CO↗
arXiv2014

History of Catalan numbers

We give a brief history of Catalan numbers, from their first discovery in the 18th century to modern times. This note will appear as an appendix in Richard Stanley's forthcoming book on Catalan numbers.

math.HO↗
arXiv2014

Growth in product replacement graphs

We prove the exponential growth of product replacement graphs for a large class of groups. Much of our effort is dedicated to the study of product replacement graphs of Grigorchuk groups, where the problem is most difficult.

math.GR↗
arXiv2014

Counting With Irrational Tiles

We introduce and study the number of tilings of unit height rectangles with irrational tiles. We prove that the class of sequences of these numbers coincides with the class of diagonals of N-rational generating functions and a class of certain binomial multisums. We then give asymptotic applications and establish connections to hypergeometric functions and Catalan numbers.

math.CO↗
arXiv2014

Lifts, derandomization, and diameters of Schreier graphs of Mealy automata

It is known that random 2-lifts of graphs give rise to expander graphs. We present a new conjectured derandomization of this construction based on certain Mealy automata. We verify that these graphs have polylogarithmic diameter, and present a class of automata for which the same is true. However, we also show that some automata in this class do not give rise to expander graphs.

math.CO↗
arXiv2014

Log-Concavity of the Partition Function

We prove that the partition function $p(n)$ is log-concave for all $n>25$. We then extend the results to resolve two related conjectures by Chen. The proofs are based on Lehmer's estimates on the remainders of the Hardy--Ramanujan and the Rademacher series for $p(n)$.

math.CO↗
arXiv2014

Bounds on the Kronecker coefficients

We present several upper and lower bounds on the Kronecker coefficients of the symmetric group. We prove $k$-stability of the Kronecker coefficients generalizing the (usual) stability, and giving a new upper bound. We prove a lower bound via the characters of $S_n$. We apply these and other results to generalize Sylvester's unimodality of the $q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in $q$: we derive explicit sharp bounds on the differences of their consecutive coefficients.

math.CO↗