Department of Computer Science, University of Otago


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1 Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly Departmet of Mathematics, Uiversity of Florida Imre Leader Departmet of Pure Mathematics ad Mathematical Statistics, Uiversity of Cambridge Status: Submitted to Aals of Combiatorics Departmet of Computer Sciece, Uiversity of Otago, PO Box 56, Duedi, Otago, New Zealad
2 PERMUTATIONS CONTAINING MANY PATTERNS M. H. ALBERT, MICAH COLEMAN, RYAN FLYNN, AND IMRE LEADER Abstract. It is show that the maximum umber of patters that ca occur i a permutatio of legth is asymptotically 2. This sigificatly improves a previous result of Colema. 1. Itroductio Give a sequece t = t 1,t 2,...,t k of distict elemets from some totally ordered set, there is a uique permutatio τ of [k] = {1, 2,...,k} with the property that for all 1 i,j k, t i < t j if ad oly if τi) < τj). We call τ the patter of t. For example, the patter of 5, 10, 2 writte i oe lie otatio is 231. I other words, the sequece represetig τ is obtaied from t simply by replacig each elemet of t by its rak i t. Let σ be a permutatio of legth, writte i oelie otatio as σ1)σ2) σ), ad thought of as a sequece of legth. For each oempty subset X of [] defie σ X to be the patter of that subsequece of σ whose idices belog to X. Defie: Pσ) = {σ X : X []}. That is, Pσ) is the set of patters that occur i σ. Also defie h) to be the maximum value of Pσ) take over all permutatios σ of legth. Trivially, h) 2 1. Slightly more precisely, for ay permutatio σ of legth : )) 1) Pσ) mi k!, k k=1 sice ot more tha k! patters of legth k ca occur. However, the expressio o the right had side of this iequality is easily see to be asymptotically 2. At the 2003 coferece o Permutatio Patters, Herb Wilf raised the issue of determiig the asymptotic) behaviour of h), ad exhibited a sequece of permutatios which established that h) exceeded the th Fiboacci umber. Micah Colema the 1
3 2 ALBERT, COLEMAN, FLYNN, AND LEADER demostrated i [1] a sequece of permutatios π, for a perfect square, 1 for which: Pπ ) > Of course this establishes that h) 1/ 2 for all, ot just perfect squares, usig the fact that h) is o decreasig). However, this left ope the questio of whether or ot h)/2 teds to 1 as teds to ifiity. I this paper, we refie the coutig argumets cocerig the umber of patters i π, for a eve perfect square, ad the exted the costructio to all other values of, i order to show that Pπ ) /2 1. Ideed, we will obtai: h) > ) 2 /2 for all positive itegers. 2. The mai costructio Let k be a positive iteger ad let = 4k 2. Let s be the sequece: s = 2k) 4k) 6k) 4k 2 ) ad cosider the permutatio π which i oe lie otatio is defied by: π = s s 1) s 2) s 2k + 1). Here s i idicates the sequece obtaied by subtractig i from each elemet of s. Geerally, we will suppress the subscript o π whe there is o risk of cofusio. Iformally, the graph of π is obtaied by takig a stadard orthogoal 2k 2k grid ad rotatig it slightly i the clockwise directio aroud its lower left had corer. We associate to each subset X of the idices of) π a 2k 2k 01 matrix, M X, whose 1 etries correspod to the elemets of the subset. We also view M X as beig partitioed ito four k k submatrices called the corer submatrices) i the usual way, that is, so that they form a 2 2 block decompositio of M X. We say that X or M X ) is ample if each k k corer submatrix of M X has o zero rows or zero colums. A example is show i Figure 1. Propositio 1. The umber of ample matrices is greater tha ) /2 1 We have adjusted the otatio slightly from that of [1] what was there called π k we are callig π k 2 so that the subscript is equal to the legth of the permutatio.
4 PERMUTATIONS CONTAINING MANY PATTERNS Figure 1. The graph of the permutatio π 64, a ample subset of its elemets idicated by filled circles, together with the correspodig matrix divided ito its corer submatrices. Proof. Recall that = 4k 2. Suppose that we sample a 01 matrix uiformly at radom from amog all 01 matrices. The probability that ay particular row or colum sum of oe of the corer submatrices is 0 is 1/2 k. There are 8k such sums which must all be o zero i order for the matrix to be ample. However, the probability that ay of them are 0 is less tha 8k/2 k. So, the probability that all are o zero is greater tha 1 8k 2 = 1 4 k 2, /2 which is equivalet to the stated result. Propositio 2. Let X ad Y be ample sets. The π X = π Y implies X = Y. Proof. We must show that, if X is ample, the it ca be recostructed from just the permutatio π X. Sice X is ample, the colum sum of both the top half ad bottom half of each colum of M X is o zero. Therefore, there are 2k 1 descets i π X, correspodig to the trasitios betwee colums of M X. Thus, we ca associate the elemets of π X with their correct colums. However, this argumet applies equally well to the rows of M X as is most easily see by cosiderig π 1. Determiig the row ad colum that represets each elemet of π X is exactly the same as recostructig X. Combiig these two results we have:
5 4 ALBERT, COLEMAN, FLYNN, AND LEADER Theorem 3. If is a eve perfect square, the h) > ) /2 We will refer to the secod term iside the paretheses above as the correctio term for this estimate. 3. Refiemets It is easy to exted the above argumets to give lower bouds o h) that are valid for all values of. We ca do this by usig the basic costructio of the previous sectio, ad addig some extra elemets i appropriate places to costruct permutatios π of legth that cotai may patters. First suppose that = 4k 2 + l where 0 < l < 2k. Take the grid associated to the permutatio π 4k 2 ad add a partial) colum o the right had side at the bottom cotaiig ot more tha k elemets, ad, if ecessary, a partial row o top at the right had side, also ot cotaiig more tha k elemets, so that the total umber of elemets added is l. As before, rotate this grid slightly, ad view the result as the graph of a permutatio, π. A example is show i Figure 2. Call the elemets of this permutatio arisig from the origial grid defiig π 4k 2 the mai elemets, ad the remaiig elemets the extra elemets. Defie a subset of the idices of π to be ample if its itersectio with the mai elemets would be ample for π 4k 2. Figure 2. The graph of the permutatio π 70, together with the matrix associated with a particular ample subset of its elemets idicated by filled circles.
6 PERMUTATIONS CONTAINING MANY PATTERNS 5 Propositio 4. Let X ad Y be ample sets. The π X = π Y implies X = Y. Proof. As before, we must describe how to recostruct X from π X. However, we ca idetify the extra elemets ad hece the mai elemets) i π X. If there are ay belogig to the ew partial colum, the they are exactly the elemets followig the 2k) th descet, while those belogig to the ew partial row, if such exist, are exactly those lyig above the maximum elemet of the first k colums. Sice the mai elemets form a ample subset of π 4k 2 we ca use the precedig result to idetify their values. Oce the values of the mai elemets are kow, so are the values of the extra elemets. Therefore, for such, h) Pπ ) > 2 4k2 1 8k ) 2 l. 2 k Certaily k /2, but also 2k + 1/2) 2 > so k > 1/2)/2. Applyig these estimates we obtai: h) > /4 ) /2 This differs from our previous estimate by a factor of 2 1/4 i the correctio term. For = 4k 2 + 2k, we switch to a grid cosistig of 2k + 1 colums of size 2k ad defie π appropriately. As i the previous sectio, we defie the four corer submatrices, except ow those o the right had side of the matrix are k k + 1) istead of k k. The probability of a subset of the matrix ot beig ample is ot as much as: 22k + 1) + 2k 2 k 2 + 2k 7k + 2 =. k 2k+1 2 k Usig the same bouds as before which still apply) plus trivial estimates for k 2 it is easy to check that the boud h) > /4 ) 2 /2 still applies i this case. We ca proceed from this poit with the halfrow/halfcolum costructio agai possibly at a pealty of aother factor of 2 1/4 i the correctio term) as far as = 2k + 1) 2. At this poit we pause for a detailed reevaluatio. I a 2k+1) 2k+1) grid, divided ito corer submatrices of sizes k k, k k + 1), k + 1) k
7 6 ALBERT, COLEMAN, FLYNN, AND LEADER ad k + 1) k + 1), the probability that a subset is ot ample is less tha: 2k k k 2 + k + 1 ) 2k + 1) + = 6k + 3. k+1 2 k 2 k+1 2 k Sice k = 1)/2, this equals 3 2) /2 We ca pursue these costructios through to the ext eve perfect square, ad, allowig for a further pealty of 2 i the correctio term which we leave to the reader to verify is geerous), obtai: Theorem 5. For all positive itegers, h) > ) /2 4. Coclusios It would be iterestig to kow just how close to 2 the value of h) actually is. A more careful aalysis of the various steps i movig from oe square grid to the ext might well provide a small improvemet i the costat factor of the correctio term of our estimate. Similarly, a aalysis of coditios weaker tha ample which oe the less would allow for a recostructio result might actually improve the asymptotic form of the correctio term. However, the simplicity of the mai costructio for = 4k 2 ) ad of the proof that ample subsets ca be recostructed from their patters, together with the lack of ay great eed for more precise estimates of h) somewhat dampes our ethusiasm for further ivestigatios i that directio. Of perhaps greater iterest would be to ivestigate the distributio of the statistic Pπ) as π rages over permutatios of legth. We would like to thak Herb Wilf for havig posed such a iterestig problem! Refereces [1] Micah Colema. A aswer to a questio by Wilf o packig distict patters i a permutatio. Electro. J. Combi., 111):Note 8, 4 pp. electroic), 2004.
8 PERMUTATIONS CONTAINING MANY PATTERNS 7 Departmet of Computer Sciece, Uiversity of Otago address: Departmet of Mathematics, Uiversity of Florida Departmet of Mathematics, Uiversity of Florida Departmet of Pure Mathematics ad Mathematical Statistics, Uiversity of Cambridge
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