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Pattern-Avoiding (0,1)-Matrices

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Pattern-Avoiding (0,1)-Matrices Nova Southeastern University NSUWorks Mathematics Faculty Articles Department of Mathematics 5-6-2020 Pattern-Avoiding (0,1)-Matrices Richard Brualdi Lei Cao Follow this and additional works at: https://nsuworks.nova.edu/math_facarticles Part of the Mathematics Commons Pattern -Avoiding (0, 1)-Matrices Richard A. Brualdi Lei Cao Department of Mathematics Department of Mathematics University of Wisconsin Nova Southeastern University Madison, WI 53706 USA Ft. Lauderdale, FL 33314 USA [email protected] [email protected] May 6, 2020 Abstract We investigate pattern-avoiding (0, 1)-matrices as generalizations of pattern-avoiding permutations. Our emphasis is on 123-avoiding and 321-avoiding patterns for which we obtain exact results as to the maximum number of 1’s such matrices can have. We also give algorithms which, when carried out in all possible ways, construct all of the pattern- avoiding matrices of these two types. Key words and phrases: permutation, pattern-avoiding, 123-avoiding, 312- avoiding, (0, 1)-matrices. Mathematics Subject Classifications: 05A05, 05A15, 15B34. arXiv:2005.00379v2 [math.CO] 5 May 2020 1 Introduction Let n ≥ 2 be a positive integer, and let Sn be the set of permutations of {1, 2,...,n}. Let k be a positive integer with k ≤ n, and let σ =(p1,p2,...,pk) be a permutation in Sk. A permutation π =(i1, i2,...,in) ∈ Sn contains the pattern σ provided there exists 1 ≤ t1 < t2 < ··· < tk ≤ n such that itr < its if and only if ptr <pts (1 ≤ r<s ≤ k); otherwise, π is called σ-avoiding; we also say that π avoids σ. Patterns in permutations and, in particular, pattern avoidance, have been extensively studied. If k = 2 and σ = (1, 2), then σ-avoiding means no ascents and the anti-identity permutation (n, . , 2, 1) is the only σ-avoiding permutation in Sn. If σ = (2, 1), then σ-avoiding means no descents and only the identity permutation (1, 2,...,n) is σ-avoiding. There is a chapter on pattern avoidance in [1]. 1 Pattern avoidance in permutations is already interesting when k = 3. Using reversal (ij goes to position n +1 − j) and complementation (ij is replaced with n +1 − ij), there are only two permutations that are inequivalent with respect to pattern avoidance: (1, 2, 3) (which is equivalent to (3, 2, 1)), and (3, 1, 2) (which is equivalent to (2, 1, 3), (1, 3, 2), and (2, 3, 1)). For each of the six permutations σ in S3 the number of σ-avoiding permutations is the Catalan number 2n C = n . n n +1 It is sometime useful to think of 1 as S(mall), 2 as M(edium), and 3 as L(arge). Then SML and LSM are the only two patterns of size 3 that need be considered. Again let σ =(p1,p2,...,pk) be a permutation in Sk. Since permutations π =(i1, i2,...,in) ∈ Sn and n × n permutation matrices are in a bijective correspondence: (i1, i2,...,in) ↔ P = [pij] where pk,ik =1(k =1, 2,...,n) and pij = 0, otherwise, σ-avoiding permutations can be studied in terms of permutation matrices. An σ-avoiding per- mutation matrix is a permutation matrix corresponding to a σ-avoiding permutation. Thus an n × n permutation matrix P is (1,2,3)-avoiding provided it does not contain as a submatrix the 3 × 3 identity matrix 1 0 0 I3 = 0 1 0 ; (1) 0 0 1 P is (3,1,2)-avoiding provided it does not contain as a submatrix the 3 × 3 matrix 0 0 1 Π3 = 1 0 0 . (2) 0 1 0 From now on we usually use the briefer notation j1j2 ··· jk for a permutation (j1, j2,...,jk) ∈ Sk. Pattern avoidance when expressed in terms of permutation matrices leads to some inter- esting questions. The n × n matrix Jn of all 1’s can be decomposed into both 123-avoiding permutation matrices and 312-avoiding permutation matrices. In fact, Jn can be decomposed into permutation matrices that are both 123-avoiding and 312-avoiding. This can be done by considering the n × n permutation matrices corresponding to the n permutations (n, n−1, n−2,..., 2, 1), (n−1, n−2,..., 2, 1, n), (n−2,..., 2, 1, n, n−1),..., (1, n, n−1,..., 2) each of which consists of an decreasing sequence followed by another decreasing sequence (empty 2 in one case). For example, 1 1 1 1 1 1 1 1 J4 = + . (3) 1 1 1 1 1 1 1 1 Recall that a permutation i1i2 ··· in is a Grassmannian provided that it contains at most one descent (so at most one consecutive 21-pattern) and is a reverse-Grassmannian provided that it contains at most one ascent (so at most one consecutive 12-pattern). The above decomposition of Jn, illustrated in (3), is a decomposition of Jn into reverse-Grassmanians. A 123-avoiding permutation need not be a Grassmannian nor a reverse-Grassmanian, for instance, 563412 is a 123-avoiding permutation but is not a Grassmannian as it has two decreases nor is it a reverse- Grassmannian as it has two increases. The notion of pattern-avoiding permutations (so pattern-avoiding permutation matrices) has been extended to general (0, 1)-matrices. Given a k × k permutation matrix Q, one seeks the largest number of 1’s in an n×n (0, 1)-matrix A such that A does not contain a k ×k submatrix Q′ such that Q ≤ Q′ (entrywise), that is, Q′ has 1’s wherever Q has 1’s but is allowed to have 1’s where Q has 0’s. Put more simply, the matrix A does not contain the pattern prescribed by Q. The emphasis here seems to have been on (asymptotic) inequalities, in particular, the F˝uredi-Hajnal conjecture [3]: Let P be any permutation matrix. Then there is a constant cP such that the number of 1’s in an n × n (0, 1)-matrix A that avoids P is bounded by cP n. Marcus and Tardos [5] solved this conjecture by proving: if P is k × k, then the number of 1’s 2 4 k in A is bounded by 2k k n. (See also pages 159–164 of [1]). There have been considerable investigations on avoiding patterns of various types in (0, 1)-matrices, again mostly in terms of asymptotic inequalities. We mention, for instance, the papers [3, 5, 6, 7, 8] and the thesis [4]. Our motivation comes from pattern avoiding in permutation matrices and the possibility of obtaining exact results for specific permutation matrices Q with equality characterizations. We focus on the two inequivalent patterns of length 3, namely 123 and 312. In the case of 123 we can more generally handle the case of the pattern 12 ··· k for arbitrary k. Also, there is no reason to assume that the matrices A are square. In the next section we consider 12 ··· k-avoiding, m × n (0, 1)-matrices. We determine the maximum number cm,n(12 ··· k) of 1’s such a matrix can contain, and characterize the matrices achieving this maximum. Surprisingly, it turns out that one can use a ‘greedy’ algorithm that always results in a 12 ··· k-avoiding, m×n (0, 1)-matrix with this maximum number cm,n(12 ··· k) of 1’s, and when carried out in all possible ways produces all such matrices. In the following section we obtain similar results for 312-avoiding matrices and its maximum number cm,n(312) of 1’s. We had hoped to settle the case of k1 ··· (k − 1) for arbitrary k, but its solution has eluded us thus far, although we are confident of the correct answer. In the final section we 3 consider some questions for future investigations. 2 12 ··· k-avoiding (0, 1)-Matrices Let A be an m × n (0, 1)-matrix. The matrix A is 12 ··· k-avoiding provided it does not contain a k × k submatrix of the form 1 ∗ ∗ ∗ ∗ ∗ 1 ∗ ∗ ∗ . .. . ∗ . ∗ ∗ (4) ∗ ∗ ··· 1 ∗ ∗ ∗ ··· ∗ 1 where ∗ denotes either a 0 or a 1. In terms of the description in the previous paragraph, Q = Ik, the k × k identity matrix. We start with the very simple case of k = 2 which contains many of the ideas used for general k. Let A = [aij] be an m × n (0, 1)-matrix with m, n ≥ 2. Then A is 12-avoiding provided that A does not have 2 × 2 submatrix of the form 1 ∗ , " ∗ 1 # where again ∗ denotes a 0 or a 1. A 12-avoiding matrix can be regarded as a generalization of a non-increasing sequence (permutation) to an arbitrary (0, 1)-matrix. The matrix A has (m + n − 1) (left-to-right) diagonals (sometimes called Toeplitz diagonals as the entries on them are constant in Toeplitz matrices) starting from any entry in its first row or first column. If A is 12-avoiding, each of these diagonals can contain at most one 1 and hence A can contain at most (m + n − 1) 1’s. We can attain (m + n − 1) 1’s with the matrix A whose first row and first column contains all 1’s and all other entries are 0. We call such a m × n (0, 1)-matrix a trivial 12-avoiding matrix. Such a matrix contains a (m − 1) × (n − 1) zero submatrix Om−1,n−1 in its lower right corner. An m × n (0, 1)-matrix A = [aij] has a complete right-to-left zigzag path, abbreviated to complete R-L zigzag path, provided (i) it has (m + n − 1) 1’s, (ii) a1n = am1 = 1, and (iii) every 1 of A except for am1 has a 1 immediately to its left or immediately below it, but not both.
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