Asymptotics of Pattern Avoidance in the Klazar Set Partition and Permutation-Tuple Settings Permutation Patterns 2017 Abstract

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1 Asymptotics of Patter Avoiace i the Klazar Set Partitio a Permutatio-Tuple Settigs Permutatio Patters 2017 Abstract Bejami Guby Departmet of Mathematics Harvar Uiversity Cambrige, Massachusetts, U.S.A. bguby@g.harvar.eu May 30, 2017 This research is joit work with Dömötör Pálvölgyi. We first itrouce set partitios a the otio of Klazar-type set partitio patter avoiace. Defiitio. A set partitio is a partitio of the set [] for some ito ay umber of oempty sets, where the orer withi the partitio is irrelevat. We call these sets blocks of the partitio. The umber of set partitios of [] is the Bell umber B. Ofte, whe writig specific set partitios, we will write the partitio [] = S 1 S k as S 1 / /S k, where the S i are i icreasig orer of smallest elemet, a are writte as strigs of umbers from least to greatest; for example, [5] = {2, 4} {1, 3, 5} woul be writte 135/24. To carry over our otios of avoiace, we efie patter cotaimet o set partitios. Defiitio. Let π a π be set partitios of [] a [m], respectively. We say that π cotais (respectively avois) π if there exists (respectively oes ot exist) a strictly icreasig fuctio f : [m] [] such that for ay i, j [m], i a j are i the same block of π if a oly if f(i) a f(j) are i the same block of π. (Note that this is istict from RGF -type set partitio avoiace as stuie i [5], where, for example, 145/23 woul avoi 12/34, as i RGF -type cotaimet the orer of blocks must be preserve.) We will be cocere with the asymptotics of patter classes of set partitios. I aalogy to the permutatio case, where all patter classes grow as! or are boue above by a expoetial, we fi that we ca similarly classify the growth rate of patter classes of set partitios to withi a expoetial factor. Specifically, we have the followig. 1

2 Theorem 1. Let P be a oempty patter class of set partitios, a as usual let P be the partitios of [] i P. The oe of the followig is true. 1. P cotais every set partitio; that is, P = B. 2. There exists a positive iteger a real costats c 2 > c 1 > 0 such that for every, c 1 (1 1 ) P c 2 (1 1 ) To explai whe the patter class falls ito a particular asymptotic rage, we relate partitios to tuples of permutatios a efie the permutability statistic. Defiitio. Let σ 1,..., σ S be permutatios. We the efie [σ 1,..., σ ] to be the set partitio of [( + 1)] cotaiig blocks B 1,..., B, where B i = {i, + σ 1 (i), 2 + σ 2 (i),..., + σ (i)}. Defiitio. Let π be a set partitio of []. The permutability of π, eote pm(π), is the miimal such that there exists m Z + a σ 1,..., σ S m such that [σ 1,..., σ ] cotais π. We ow have the termiology to escribe which patter classes correspo to which i Theorem 1: is the smallest permutability of a set partitio ot i P. (We igore the case where this is 0 this correspos also to = 1.) We ow outlie the methos of proof of the lower a upper bous. For the lower bou, ote that by assumptio, all partitios of permutability 1 are i P. This will iclue all partitios of the form [σ 1,..., σ 1 ]. O [], there are (!) 1 such partitios, which alreay gives the esire lower bou. As usual, the upper bou is far more ifficult; we will simply escribe some of the techiques a lemmas ivolve. It suffices to show the result for classes give by avoiig oe elemet [σ 1,..., σ ], as every patter class with the correct asymptotic is cotaie i such a class. As with permutatios, where ofte it is ecessary to geeralize to 0 1-matrices, we may geeralize from set partitios to orere hypergraphs. Just as 0 1 matrices correspo to bipartite graphs, with permutatio matrices correspoig to matchigs or 1-regular bipartite graphs, set partitios correspo to 1-regular orere hypergraphs. I particular, the eges of a 1-regular orere hypergraph give a partitio of the vertex set []. This implies that we shoul look for some Fürei-Hajal type bou o hypergraphs that avoi a hypergraph of the esire form. We first make several efiitios. Defiitio. The orere hypergraph G cotais (respectively avois) the orere hypergraph H if there exists (respectively oes ot exist) a orer-preservig ijectio i V : V (H) V (G) a a a ijectio i E : E(H) E(G) that are compatible, i the sese that if v e E(H), the i V (v) i E (e). A -permutatio hypergraph is a hypergraph correspoig to some set partitio [σ 1,..., σ 1 ]. 2

3 Note that the efiitio esures that a -permutatio hypergraph is -uiform. We may ow state our Fürei-Hajal type lemma. Lemma 1. For a orere hypergraph G, efie i(g) := E. Let H be a -permutatio E E(G) hypergraph. The there exists a costat c such that for all Z + a orere hypergraphs G o [] avoiig H, i(g) c 1. The proof of the lemma is a techical iuctio, first provig a stroger statemet for t-uiform hypergraphs a the usig that to obtai the result for geeral graphs. Note that Lemma 1 is a geeralizatio of two results of Klazar a Marcus i [4], oe of which is simply the = 2 case of Lemma 1, a the other eals with -imesioal 0 1 matrices, which give a subset of the cases whe G is -uiform. Remark. This result is quite iterestig i its ow right, a prompts the questio of etermiig asymptotics of i(g) for various H (or to geeralize to patter classes of max G avois H V (G)=[] orere hypergraphs). From Lemma 1 it is ot ifficult to see that if H is 1-regular, correspoig to some set partitio π, the this is withi a costat o either sie of pm(π). For o-1-regular hypergraphs, the aswer may be more complicate; for example, if G is restricte to be a graph, Klazar shows i [3] that it is possible to achieve a fuctio that grows very slightly faster tha liearly. From Lemma 1 it is a routie recursive erivatio that the umber of orere hypergraphs G o [] that avoi some -permutatio hypergraph H is boue above asymptotically by c 1 for some costat c. Ufortuately, this is far greater tha the esire (1 1) 1, as we are ow cosierig geeral hypergraphs a ot just set partitios. However, a fial trick fiishes the solutio; give a set partitio i the form of a 1-regular hypergraph G o [], we may ivie the vertices ito s itervals of size. For each ege s E of the origial graph, we ca create a ege E of the ew graph cotaiig exactly those vertices correspoig to itervals i which E cotaie at least oe vertex. (We the elete uplicate eges.) If the resultig graph is G, we kow that we oly have c s 1 choices for G. The, we may bou how may set partitios correspo to a particular choice of G. Optimizig for s leas after some algebra to the esire bou. We may ow apply this result to permutatio-tuple patter avoiace, which we ow efie. Defiitio. Fix Z +. Let σ 1,..., σ S a σ 1,..., σ S m be permutatios. The the -tuple (σ 1,..., σ ) cotais (respectively avois) the -tuple (σ 1,..., σ ) if there exist (respectively o ot exist) iices i 1,..., i m [] with i 1 < < i m satisfyig the property that for ay j, σ j (i 1 ) σ j (i m ) has the same relative orerig as σ j(1) σ j(m). I other wors, a -tuple T 1 cotais aother -tuple T 2 if a oly if each permutatio i T 1 cotais the correspoig permutatio i T 2 at the same locatio. 3

4 It is ot ifficult to see that -permutatio-tuple avoiace is equivalet to set partitio avoiace where the set partitios are restricte to the form [σ 1,..., σ ]. That is, [σ 1,..., σ ] cotais [σ 1,..., σ ] if a oly if (σ 1,..., σ ) cotais (σ 1,..., σ ). This immeiately gives a upper bou o the umber of elemets of S that avoi a particular -tuple (σ 1,..., σ ): it is at most the umber of set partitios of [( + 1)] that ( avoi [σ 1,..., σ ]. By our earlier result this is at most c for some costat c. Itriguigly, the lower bou i this case also hols (this follows quickly from results i [1]) that is, the umber of elemets of S that avoi (σ 1,..., σ ) is at least c some costat c > 0. I sum, we have the followig result. ( for Theorem 2. Fix, m Z +, a σ 1,..., σ S m. For a particular, let S (σ 1,..., σ ) be the set of -tuples of permutatios i S that avoi (σ 1,..., σ ). The there exist costats c 2 > c 1 > 0 so that for all, c 1 ( S (σ 1,..., σ ) < c 2 ( 2 1 Whe = 1, Theorem 2 simply returs Staley-Wilf. For higher this is highly otrivial eve i simple cases for example, there oes ot seem to be a close form for the umber of pairs of permutatios i S that avoi (12, 12). Eve the asymptotics here are ot well boue: by Theorem 2 we kow that it will be boue withi a expoetial of (!) 3 2 (ote that i ay of these theorems, by Stirlig Approximatio, we ca replace by!), but the base of that expoetial factor is oly curretly kow (to the author s kowlege) to be boue betwee 1 a 3 3(3 log 3 4 log 2) 3.76, as prove i [2]. (This special case is the sequece of umber of pairs (σ 1, σ 2 ) S 2 with σ 1 σ 2 i the weak Bruhat orer.) There are several questios that remai here; most immeiately, i the case of -tuples, we have ot show ay classificatio of growth rates of patter classes like the oe we showe i the set partitio case. Of course, the patter class of every -tuple (that is, basis size 0) grows as (!) (or, to use the kis of expressios above, withi expoetially of ); Theorem 2 shows that ay patter class of -tuples of permutatios with exactly oe ( (otrivial) basis elemet grows withi expoetially of. Other growth orers are possible; we ca take the prouct of a patter class of 1 -tuples a a patter class of 2 - tuples to get a patter class of tuples. This immeiately implies that we may obtai k growth rates withi expoetial of c 1 where c =, where k, 1,..., k Z + with i=1 i k. Are there ay other posible growth rates? It is ot clear what we might expect from, for example, a arbitrary patter class of pairs of permutatios (so = 2) with basis size 2. The permutability statistic of partitios oes ot seem to be istribute accorig to other kow statistics of partitios. Determiig this istributio might help specify the costats i Theorem 1, as P is boue below by the umber of set partitios of [] permutability at most 1. ). 4

5 Refereces [1] Graham Brightwell, Raom k-dimesioal Orers: With a Number of Liear Extesios, Orer, Vol. 9, Iss. 4, pp , [2] Ewar Crae a Nic Georgiou, Notes o Atichais i the Raom k-dimesioal Orer, Bristol Workshop o Raom Atichais i k-dimesioal Partial Orers (2011), receive via persoal commuicatio with Aam Hammett 8/18/2016. [3] Marti Klazar, Extremal Problems for Orere (Hyper)graphs: Applicatios of Daveport-Schizel Sequeces, Europea Joural of Combiatorics, Vol. 25, Iss. 1, pp , [4] Marti Klazar a Aam Marcus, Extesios of the Liear Bou i the Fürei-Hajal Cojecture, Avaces i Applie Mathematics, Vol. 38, Iss. 2, pp , [5] Toufik Masour, Combiatorics of Set Partitios, CRC Press,