For a permutation matrix $P$, let $s_P$ denote its Stanley-Wilf limit, the exponential growth rate of the number of $n\times n$ permutation matrices avoiding $P$. Let $c_P$ denote its Füredi-Hajnal limit, which is the limit $\displaystyle \lim_{n \to \infty} \text{ex}(n,P)/n$ where $\text{ex}(n,P)$ is the maximum number of ones in an $n\times n$ $0$-$1$ matrix avoiding $P$. Cibulka proved the universal quadratic bound $s_P\leq 2.88\,c_P^2$. In this note we improve the constants in Cibulka's result through a so-called ``block contraction" argument. Defining \[ F(c)=\inf_{t\in\mathbb{N}} \frac{(t!)^{1/t}\,15^{\,c/t}}{c}, \] for $c>0$, this leads us to the revised inequality $s_P\leq F(c_P)\,c_P^2$. In particular, $F(c)=\log 15+o(1) \approx 2.70805\ldots +o(1)$ as $c\to\infty$, and the constant improves $2.88$ once $c_P \geq 17$.
Let S n be the symmetric group on [n] = {1, . . . , n}. For π ∈ S k and σ ∈ S n , we say that σ contains π if there exist indices 1 ≤ i 1 < i 2 < • • • < i k ≤ n such that for every 1 ≤ a, b ≤ k, one has σ ia < σ i b if and only if π a < π b . Otherwise σ avoids π. We write Av n (π) = {σ ∈ S n : σ avoids π}.
The Stanley-Wilf conjecture asserted that for each fixed π the sequence | Av n (π)| is bounded above by C n for some constant C = C(π). The conjecture was proved by Marcus and Tardos via a strengthening in the setting of forbidden submatrices [5].
There is a convenient matrix formulation of this Stanley-Wilf limit. Let P be the k × k permutation matrix of π. An n × n 0-1 matrix A contains P if there are row indices r 1 < • • • < r k and column indices c 1 < • • • < c k such that A r i ,c j = 1 whenever P i,j = 1. Equivalently, A contains a k × k submatrix that dominates P entrywise. If no such choice of rows and columns exists, we say that A avoids P . Let S P (n) be the number of n × n permutation matrices avoiding P . Then S P (n) = | Av n (π)|, and we set
which is the Stanley-Wilf limit of P . In this light, permutation patterns may be formulated either in terms of permutations or in terms of their permutation matrices. We will move freely between these viewpoints, since the main estimate is most transparent on matrices. Relatedly, the forbidden submatrix problem asks for the maximum number of ones in an n × n 0-1 matrix avoiding a fixed pattern. Following Füredi and Hajnal [3], define ex(n, P ) = max{|A| : A ∈ {0, 1} n×n avoids P }, where |A| denotes the number of ones in A. For permutation matrices P , Marcus and Tardos proved the linear bound ex(n, P ) = O(n) [5]. It is convenient to record this linear behavior in the normalized limit
usually called the Füredi-Hajnal limit of P . The relationship between s P and c P has been studied extensively. Klazar showed that linear bounds for ex(n, P ) imply exponential bounds for S P (n) by a contraction argument [4]. Cibulka sharpened this reduction and proved a universal quadratic inequality s P ≤ 2.88 c 2 P for all permutation matrices P [2]. The purpose of this note is to strengthen the constant in Cibulka’s result to F (c P ) which interpolates between the explicit constant 2.88 and the asymptotic constant log 15 ≈ 2.70805 . . .. The number 15 comes from a two-by-two block contraction in which each nonzero block has 2 4 -1 = 15 possible 0-1 patterns. This is the basis of Klazar’s argument [4] for a coarser version of Cibulka’s inequality. Our argument for strengthening the constants comes from a similar process block contraction argument, but with larger block sizes.
Then for every permutation matrix P one has
The function F (c) can be evaluated numerically rather quickly, because the infimum is attained for t near c log 15. In particular the bound already improves the constant 2.88 once c ≥ 17. For larger c, one may fix a choice such as t = 3c and combine it with Stirling’s formula to obtain an explicit uniform bound below 2.88. Concretely, Stirling’s upper bound gives (t!) 1/t ≤ (t/e)(2πt) 1/(2t) e 1/(12t 2 ) , and with t = 3c this yields
Let G(c) be the function on the right hand side. Since G(c) > 0 we can take logarithmic derivatives and multiply by 6c 3 > 0 to get
For c ≥ 1 we have log(6πc) > 1, and therefore (log G(c)) ′ < 0, so G is decreasing. Furthermore explicit computation reveals G(18) < 2.88. The result then follows. □
In Theorem 1, we shall see that F (c) = log(15) + o(1) ≈ 2.70805 + o(1) as c → ∞. In Corollary 2, we see that F (c) decreases in c for positive integers, so we expect F (c) to interpolate between just below 2.88 and 2.70805 for c ≥ 17.
We now describe the block contraction argument, together with the two estimates that feed into it. The first is an upper bound on the number of lifts of a fixed contracted matrix, and the second combines this with an upper bound on the number of contracted matrices in terms of ex(n, P ).
Fix integers t, n ≥ 1 and set N = tn. Partition [N ] into consecutive intervals I 1 , . . . , I n of length t. For an N × N 0-1 matrix A, define its t-contraction cont t (A) to be the n × n 0-1 matrix B given by B a,b = 1 if and only if A has a one in the block I a × I b .
When A is a permutation matrix, the contraction records which row blocks meet which column blocks.
A basic point is that contraction preserves pattern avoidance for permutation patterns. If cont t (A) contains a permutation matrix P , then by definition there are k row blocks and k column blocks in increasing order so that each of the k prescribed blocks contains at least one one-entry of A. Because the row blocks are disjoint and the column blocks are disjoint, selecting one witness one-entry from each prescribed block produces a copy of P in A. Thus, a permutation matrix that avoids P always contracts to a 0-1 matrix that avoids P .
For an n × n 0-1 matrix B, let L t (B) be the set of N × N permutation matrices A whose suppo
This content is AI-processed based on open access ArXiv data.