Permutation classes and infinite antichains

Transcription

1 Permutation classes and infinite antichains Robert Brignall Based on joint work with David Bevan and Nik Ruškuc Dartmouth College, 12th July 2018

2 Typical questions in PP For a permutation class C: What is the growth rate? What is the generating function? (e.g. rational, algebraic, D-finite) What is the basis? (Is it finite?) What do the permutations look like?

3 Examples: Av(231) and Av(321) Both enumerated by Catalan numbers: f (z) = 1 1 4z 2z Av(231) Av(321) Growth rate 4 4 Generating function algebraic algebraic Basis Look like Av(231) Av(231)

4 What about subclasses of Av(231), Av(321)? Growth rate C Av(231) D Av(321) Generating function Basis

5 What about subclasses of Av(231), Av(321)? Growth rate Generating function C Av(231) Countably many possibilities D Av(321) Includes [2.36, 2.48] (Bevan, 2018) Basis

6 What about subclasses of Av(231), Av(321)? Growth rate Generating function Basis C Av(231) Countably many possibilities Rational (Albert, Atkinson, 2005) D Av(321) Includes [2.36, 2.48] (Bevan, 2018) Could be anything

7 What about subclasses of Av(231), Av(321)? Growth rate Generating function C Av(231) Countably many possibilities Rational (Albert, Atkinson, 2005) D Av(321) Includes [2.36, 2.48] (Bevan, 2018) Could be anything Basis Finite Finite or infinite What s causing Av(321) to misbehave?

8 1 An antichain is born

9 Back in Sorting Using Networks of Queues and Stacks ROBERT TAR JAN Stanford University,* Stanford, California AI~STRAC'r. The problem of sorting a sequence of numbers using a network of queues and stacks is presented. A characterization of sequences sortable using parallel queues is given, and partial characterizations of sequences sortable using parallel stacks and networks of queues are given. KEY WORDS AND PHRASES: sorting, network, queue, stack CR CATEGORIES: 5.31, 5.32 Inspired by Knuth [2, p. 234], we wish to consider the following problem: Suppose we are presented with the layout of a railroad switchyard (Figure 1 ). If a train is driven into one end of the yard, what rearrangements of the cars may be made before the train comes out the other end? In order to get a handle on the problem, we must introduce some formalization. A switchyard is an acyclic directed graph, with a unique source and a unique sink

10 An antichain is born

11 An antichain is born

12 An antichain is born

13 An antichain is born The increasing oscillating antichain, Osc, avoids 321

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16 Formality, briefly An antichain: any set of permutations where no permutation is contained in any other. A = {α 1, α 2,... : α i α j for all i, j}. N.B. By minimality, the basis of a permutation class is always an antichain. Well-quasi-order or partial well-order: no infinite antichains. [I ll avoid these terms in this talk.]

17 Antichains, bases and enumeration Proposition (Atkinson, Murphy, Ruškuc, 2002) Let C be a finitely based permutation class. The following are equivalent: (1) Every subclass of C is finitely based, (2) C contains at most countably many subclasses, (3) C has no infinite antichain. Conjecture (Vatter, 2015) If C is a permutation class that contains no infinite antichains, then it has an algebraic generating function.

18 Back to those subclasses of Av(231), Av(321) Growth rate Generating function C Av(231) Countably many possibilities Rational (Albert, Atkinson, 2005) D Av(321) Includes [2.36, 2.48] (Bevan, 2018) Could be anything Basis Finite Finite or infinite Infinite antichains None Osc

19 Diversion 1: intervals of growth rates Theorem (Bevan, 2018; Vatter, 2010) Every real number above is the growth rate of some permutation class. Proof. Create sum-closed classes by choosing sum indecomposables from Osc, and some easy variants. Using cleverness, find a class with any growth rate above the unique real root of x 8 2x 7 x 5 x 4 2x 3 2x 2 x 1. N.B. sum-closed guarantees existence of a growth rate (Arratia, 1999)

20 Underpinning Osc Osc (above) and its easy variants all build on increasing oscillations: N.B. These form a chain (not an antichain!).

21 a permutation r, determining the shortest product of generators that equals r is NPhard. Jerrum [J85] proves that the problem is PSPACE-complete, and remains so, Diversion when restricted2: to two Gollan generators. permutations In our problem, the generator set is fixed. However, Kececioglu and Sankoff [KS93] conjecture that sorting by reversals is NP-complete. Gates and Papadimitriou [GP79] studied a similar sorting by prefix reversals problem (also known as pancake-flipping problem)" given an arbitrary permutation, find SIAM J. COMPUT0 () 1996 Society for Industrial and Applied Mathematics Vol. dpre,(r), 25, No. 2, which pp , is theapril minimum 1996 number of reversals of the form p(1, i) that sort OO3 Their concern is with bounds on the prefix reversal diameter of the symmetric group, dpre$(n) maxrs dpre$(tr). They show that dpre.f(n) <_ n + (see also [GT78]) and that for infinitely many n, dpref(n) >_ n17 [GP79]. Aigner and West [AW87] consider GENOME the diameter REARRANGEMENTS of sorting when the operation AND SORTING is reinsertion BYofREVERSALS* the first element, and Amato et al. [ABSR89] VINEET consider BAFNAt aand variation PAVELinspired A. PEVZNER$ by reversing trains. Kececioglu and Sankoff [KS93] found an approximation algorithm for sorting by reversals Abstract. Sequence comparison in molecular biology is in the beginning of a with performance guarantee 2. They also devised efficient bounds, allowing major paradigm them to shift--a shift from gene comparison based on local mutations (i.e., insertions, deletions, and substitutions ofthe nucleotides) solve reversal to distance chromosome problem comparison optimally based or on almost global rearrangements optimally for (i.e., n ranging from inversions and transpositions 30 to 50. This of range fragments). covers The the classical biologically methodsimportant of sequencecase comparison of mitochondrial do not workgenomes. for global rearrangements, Define d(n) and little maxes is known d(r) in computer to be thescience reversal about diameter the edit distance of the symmetric between sequences group if ofglobal orderrearrangements n. Gollan conjectured are allowed. that In thed(n) simplestn- form, 1 and the problem that only of gene onerearrangements permutation corresponds to sorting by reversals, i.e., sorting of an array using reversals of arbitrary Vn, and its inverse, VI, require n- 1 reversals to be sorted (see Kececioglu fragments. and Recently, Kececioglu and Sankoff gave the first approximation algorithm for sorting by reversals with guaranteed Sankoff [KS93] error bound for details). 2 and identified The Gollan open problems permutation, related toin chromosome one-line notation, rearrangements. is defined One of asthese follows: problems is Gollan s conjecture on the reversal diameter of the symmetric group. This paper proves the conjecture. Further, the problem. of expected reversal distance between two random permutations is investigated. (3, 1,5,2,7,4,...,n- The reversal 3, distance n- 5, between n- 1, n- two 4, random n,n-permutations 2), n even, is shown to be very7nclose to the reversal diameter, thereby indicating that reversal distance provides a good separation between (3, 1, 5, 2, related and 7, 4, n 6, n 2, n nonrelated sequences in molecular 5, n, n 3, n evolution 1), n odd. studies. The gene rearrangement problem forces us to consider reversals of signed permutations, as the genes in DNA could be positively For n < or negatively 11, Gollan oriented. verified Anthis approximation conjecturealgorithm using extensive for signed permutation computations. is presented, Kececiogluprovides and Sankoff a performance [KS93] developed guarantee of lower Finally, bounds using forthe reversal signed distance, permutations allowing approach, them which an

22 Diversion 3: Rational superclasses Theorem (Albert, B., Vatter, 2013) Every proper permutation class C is contained in a permutation class with a rational generating function. Proof. Use increasing oscillations to make an enormous infinite antichain

23 Diversion 3: Rational superclasses Theorem (Albert, B., Vatter, 2013) Every proper permutation class C is contained in a permutation class with a rational generating function. Proof. Use increasing oscillations to make an enormous infinite antichain A

24 Diversion 3: Rational superclasses Theorem (Albert, B., Vatter, 2013) Every proper permutation class C is contained in a permutation class with a rational generating function. Proof. Use increasing oscillations to make an enormous infinite antichain A such that Av(A) A has a rational generating function. Union this with C, and remove enough antichain elements of each length to preserve rationality.

25 2 Labelled containment

26 Labelled containment Colour/label each entry of σ and π red or black. σ l π if σ embeds in π so that the labels match up. (Generalisations with more labels possible.) Examples: l l

27 Labelled containment and Osc Increasing oscillations only embed contiguously

28 Labelled containment and Osc Increasing oscillations only embed contiguously

29 Labelled containment and Osc Increasing oscillations only embed contiguously

30 Labelled containment and Osc Increasing oscillations only embed contiguously

31 Labelled containment and Osc l Cannot embed lowest & highest into lowest & highest

32 A labelled infinite antichain Warning! Abuse of notation: really means A class C contains a labelled infinite antichain C contains an infinite set of permutations whose entries can be labelled red/black so that it forms an infinite antichain in l.

33 No labelled antichain finite basis Proposition (After Pouzet, 1972) A permutation class C that contains no infinite labelled antichain is finitely based. Proof. Suppose C = Av(B) is not finitely based. For each β B: β

34 No labelled antichain finite basis Proposition (After Pouzet, 1972) A permutation class C that contains no infinite labelled antichain is finitely based. Proof. Suppose C = Av(B) is not finitely based. For each β B: β

35 No labelled antichain finite basis Proposition (After Pouzet, 1972) A permutation class C that contains no infinite labelled antichain is finitely based. Proof. Suppose C = Av(B) is not finitely based. For each β B: β β B = {β : β B} is a labelled antichain in C: contradiction.

36 Back to those subclasses of Av(321) Theorem (Albert, B., Ruškuc, Vatter) If D Av(321) is finitely based or does not contain an infinite antichain, then it has a rational generating function. Furthermore, if D Av(321) contains a (labelled or unlabelled) infinite antichain then it contains long increasing oscillations. Thus, if D Av(321) avoids some oscillation, then D is finitely based and has a rational generating function.

37 Small permutation classes What growth rates are allowed? Kaiser, Klazar, 2003 Vatter, 11 Bevan 18, Vatter φ 2 κ λ Growth rates below κ : no (labelled or unlabelled) infinite antichains ( all classes finitely based). At κ: increasing oscillations and Osc appear ( uncountably many classes). Theorem (Albert, Ruškuc, Vatter, 2015) Every permutation class C with gr(c) < κ has a rational generating function.

38 Smallish permutation classes ξ marks another phase transition: from countably many to uncountably many different growth rates (Vatter). Classification of growth rates from κ to ξ (Pantone, Vatter).

39 Smallish permutation classes ξ marks another phase transition: from countably many to uncountably many different growth rates (Vatter). Classification of growth rates from κ to ξ (Pantone, Vatter). Conjecture The next infinite (labelled) antichain first appears in a class of growth rate ν

40 Smallish permutation classes ξ marks another phase transition: from countably many to uncountably many different growth rates (Vatter). Classification of growth rates from κ to ξ (Pantone, Vatter). Conjecture The next infinite (labelled) antichain first appears in a class of growth rate ν Conjollary Any class C with gr(c) < which contains only bounded length oscillations is finitely based.

41 3 Grid classes

42 Grid classes M a 0, ±1 matrix. π Grid(M) if π can be gridded so that each cell of π is empty increasing decreasing if the corresponding entry of M is M = ( 1 1 ) π =

43 Grid classes M a 0, ±1 matrix. π Grid(M) if π can be gridded so that each cell of π is empty increasing decreasing if the corresponding entry of M is M = ( 1 1 ) Grid(M) π =

44 The antithesis of Osc Long increasing oscillations don t grid nicely.

45 The antithesis of Osc Long increasing oscillations don t grid nicely.

46 Those questions again Growth rate Grid(M) Generating function Basis Infinite antichains

47 Those questions again Grid(M) Growth rate ρ(m) 2 (Bevan, 2015) Generating function Basis Infinite antichains

48 Those questions again Grid(M) Growth rate ρ(m) 2 (Bevan, 2015) Generating function Rational if acyclic, otherwise... Basis Infinite antichains see Albert, Atkinson, Bouvel, Ruškuc, Vatter, 2013

49 Those questions again Grid(M) Growth rate ρ(m) 2 (Bevan, 2015) Generating function Rational if acyclic, otherwise... Basis Finite if acyclic, otherwise... Infinite antichains see Albert, Atkinson, Bouvel, Ruškuc, Vatter, 2013

50 Those questions again Grid(M) Growth rate ρ(m) 2 (Bevan, 2015) Generating function Rational if acyclic, otherwise... Basis Finite if acyclic, otherwise... Infinite antichains None iff acyclic (Murphy, Vatter, 2003) see Albert, Atkinson, Bouvel, Ruškuc, Vatter, 2013

51 Cycles in a grid class Example M: m n matrix. G M : bipartite graph, vertices {c 1,..., c m, r 1,..., r n }, with c i r j E(G M ) if M ij = 0. M = ( 1 1 ) G M : r 1 r c 1 c 2 c 3

52 Cycles in a grid class Example M: m n matrix. G M : bipartite graph, vertices {c 1,..., c m, r 1,..., r n }, with c i r j E(G M ) if M ij = 0. M = ( 1 1 ) G M : r 1 r c 1 c 2 c 3

53 Cycles in a grid class Example M: m n matrix. G M : bipartite graph, vertices {c 1,..., c m, r 1,..., r n }, with c i r j E(G M ) if M ij = 0. M = ( 1 1 ) G M : r 1 r c 1 c 2 c 3

54 Cycles in a grid class Example M: m n matrix. G M : bipartite graph, vertices {c 1,..., c m, r 1,..., r n }, with c i r j E(G M ) if M ij = 0. M = ( 1 1 ) G M : r 1 r c 1 c 2 c 3

55 Cycles in a grid class Example M: m n matrix. G M : bipartite graph, vertices {c 1,..., c m, r 1,..., r n }, with c i r j E(G M ) if M ij = 0. M = ( 1 1 ) G M : r 1 r 2 1 c 1 c 2 c 3

56 Cycles in a grid class Example M: m n matrix. G M : bipartite graph, vertices {c 1,..., c m, r 1,..., r n }, with c i r j E(G M ) if M ij = 0. M = ( 1 1 ) G M : r 1 r 2 c 1 c 2 c 3 Acyclic: G M has no cycles. Unicyclic: G M has at most one cycle.

57 Questions by cyclicity Grid(M) acyclic unicyclic polycyclic Growth rate ρ(m) 2 ρ(m) 2 ρ(m) 2 Generating function Rational Erm... Erm... Basis Finite Erm... Erm... Infinite antichains None Some... Some... Not even labelled infinite antichains ( finitely based).

58 Unicyclic grids Following the role of Osc in Av(321), we ask: Question When does a subclass C of a unicyclic grid class Grid(M) contain an infinite labelled antichain?

59 Key idea: -decomposition Example: Let M =

60 Key idea: -decomposition Example: Let M =

61 Key idea: -decomposition Example: Let M = =

62 Key idea: -decomposition Example: Let M = =

63 Key idea: -decomposition Example: Let M = =

64 Key idea: -decomposition Example: Let M = = Lemma The class C Grid(M) contains infinite labelled antichains if and only if the -indivisible gridded permutations in C do.

65 Consequences Theorem (Bevan, B., Ruškuc) For a unicyclic grid class Grid(M), any subclass C Grid(M) contains (labelled) infinite antichains if and only if C has infinite intersection with one of two labelled antichains (per component of G M ) Example: For M = 0 1 1, they are: 1 1 0

66 Basis of unicyclic grids Theorem (Bevan, B., Ruškuc) Unicyclic grid classes are finitely based. Proof outline (if it survives writing up... ) If C Grid(M) contains no labelled antichains, then neither does C +1.

67 Basis of unicyclic grids Theorem (Bevan, B., Ruškuc) Unicyclic grid classes are finitely based. Proof outline (if it survives writing up... ) If C Grid(M) contains no labelled antichains, then neither does C +1. Write Grid(M) = Av(B) and argue that B C +1.

68 Enumeration of unicycles Proposition (Bevan, PhD thesis 2015) The gridded permutations in a unicyclic grid class have an algebraic generating function. Work in progress A unicyclic grid class should have an algebraic generating function.

69 Questions by cyclicity Grid(M) acyclic unicyclic polycyclic Growth rate ρ(m) 2 ρ(m) 2 ρ(m) 2 Generating function Rational Erm... Erm... Basis Finite Erm... Erm... Infinite antichains None Some... Some... Not even labelled infinite antichains ( finitely based).

70 Questions by cyclicity Grid(M) acyclic unicyclic polycyclic Growth rate ρ(m) 2 ρ(m) 2 ρ(m) 2 Generating function Rational?Algebraic Erm... Basis Finite Finite Erm... Infinite antichains None Two Some... Not even labelled infinite antichains ( finitely based).

71 Thanks!