lattice paths from an order-theoretic perspective · 2016-06-07 · lattice paths from an...

Lattice paths from an order-theoretic perspective

L. Ferrari

Dipartimento di Matematica e Informatica, Universita degli Studi di Firenze,Viale Morgagni 65, 50134 Firenze, Italy

[email protected]

GASCom 2016,Bastia, 2-4 June 2016

Outline

Table of contents IIntroduction

PathsPosetsOutline of the talk

Lattices of paths

Distributive lattice structureRepresentationsThe Euler Characteristic

Unimodality

Heyting algebra structureDyck algebrasThe logic of subintervalsPosets of intervalsOpen problems

Counting chains in lattices of pathsChains in Dyck latticesSaturated chains of length 1

Outline

Table of contents IISaturated chains in Dyck latticesOpen problems

Schroder partitions and Schroder tableauxThe case of Schroder pathsThe poset of Schroder partitionsAn RS-like correspondence for Schroder tableauxAn alternative presentation of Schroder tableauxFurther work

Isomorphisms with other combinatorial posetsDyck latticesMotzkin (and Schroder) latticesGrand-Dyck lattices

The notion of path patternThe Dyck pattern poset(Principal) pattern avoiding classesThe Mobius functionOpen problems

Introduction

A class of paths is the set of all paths starting at the origin of a fixedCartesian coordinate system, ending on the x-axis, never going below thex-axis, and using only a prescribed set of steps.

E.g.: Dyck paths tU,Du.

Motzkin paths tU,H,Du.

Schroder paths tU,H2,Du.

Introduction

A class of paths is the set of all paths starting at the origin of a fixedCartesian coordinate system, ending on the x-axis, never going below thex-axis, and using only a prescribed set of steps.

E.g.: Dyck paths tU,Du.

Motzkin paths tU,H,Du.

Schroder paths tU,H2,Du.

Introduction

Grand paths

Given a class of paths, the associated class of Grand paths is defined inthe same way, except for the fact that we are allowed to go below thex-axis.

E.g.: Grand Dyck paths

Grand Motzkin paths

Grand Schroder paths

Introduction

Grand paths

Given a class of paths, the associated class of Grand paths is defined inthe same way, except for the fact that we are allowed to go below thex-axis.

E.g.: Grand Dyck paths

Grand Motzkin paths

Grand Schroder paths

Introduction

Posets

Posets of paths

Any class of (Grand) paths of the same length can be endowed with anatural partial order structure, by declaring P ¤ Q when P lies weaklybelow Q in the usual two-dimensional drawing of paths.

Below are depicted the Hasse diagrams of the lattices of Dyck paths ofsemilength 3, Motzkin paths of length 4 and Schroder paths ofsemilength 2, respectively.

D3 M4 S2

Introduction

Posets

Posets of paths

D3 M4 S2

Introduction

Posets

Posets of paths

D3 M4 S2

Introduction

Posets

Posets of pathsIn some cases, this partial order has already appeared in some differentform. For instance, in the Dyck case there is an alternative description interms of Young lattices (precisely, Dyck posets are order-isomorphic tothe Young lattices of staircase partitions).

Another instance of these posets (which are actually lattices) appears ina paper by Stanley (1975), and for this reason they are called Stanleylattices by Bernardi (2009). Moreover, Cautis and Jackson (2003) find anappearance of Dyck posets in the study of the matrix of chromatic joins.

In the last decade, Dyck posets have been investigated (directly orindirectly) by several authors (Sapounakis, Tasoulas, Tsikouras (2006),Santocanale (2007), Owczarek, Prellberg (2012), Barnabei, Bonetti,Silimbani (2013), Braun, Browder, Klee (2013), Muhle (2013,2014,2015),Blanco, Petersen (2014), Gobet, Williams (2016),...).

However, it does not seem that this approach has ever been tried forlattice paths in general. At any rate, I believe that the language of latticepaths gives a geometric flavor to the subject which allows to expressseveral properties in a more fascinating way.

Introduction

Posets

Introduction

Posets

Introduction

Posets

Introduction

Outline of the talk

(Possibly) in this talk...Some natural and interesting problems concerned with this partial orderstructure are the following.

� In which cases do we get lattices?� In case we obtain (finite) distributive lattices:

� what can we say on the poset of their join-irreducibles?(ù (Birkhoff’s) representation theorems);

� Euler characteristic;� rank-unimodality (even in case we just have ranked posets, of

course);� Heyting algebra structure and logic-theoretic applications;

� Enumeration of (saturated) chains.� enumeration of edges (= saturated chains of length 1) in several

lattices of paths;� enumeration of chains and saturated chains in Dyck lattices;� new structures for counting chains in Schroder lattices: Schroder

tableaux.

� Isomorphisms with other interesting posets having combinatorialrelevance.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

tableaux.

Introduction

Outline of the talk

(Possibly) in this talk...

Another interesting issue of an order-theoretic nature concerns a different(although equally natural) partial order relation defined on the set of allpaths of a certain class, which is the pattern containment relation.

This partial order is defined in analogy with the classical patterncontainment order on permutations.

Here we will have a look at some structural and enumerative propertiesof the pattern containment order in the special case of Dyck paths.

Introduction

Outline of the talk

Introduction

Outline of the talk

Introduction

Outline of the talk

Not in this talk...

We point out that there are other instances of partial orders, more or lessnaturally defined on lattice paths, that we are not going to consider here.

Among them, the most well known are certainly the Tamari lattices,widely studied by many authors, but probably better espressed in terms ofdifferent combinatorial structures.

A little bit less known, but not less interesting, are also other types oflattices considered by Baril and Pallo (2006,2008,2014), such as thephagocyte lattices and the pruning-grafting lattices.

Introduction

Outline of the talk

Not in this talk...

Introduction

Outline of the talk

Not in this talk...

Lattices of paths

Distributivity

In which cases do we get lattices? We still don’t know in general...

Simpler question: in which cases do we get distributive lattices withcoordinatewise meet and join? We have a characterization, when theclass of paths only has unitary steps.

Unitary step means: each step is of type p1, kq, for some k P Z. A class ofpaths of length n having only unitary steps can be identified with the set

CΓn � tf : r0, ns Ñ N | f p0q � f pnq � 0, f pk � 1q � f pkq P Γ,@k nu,

where Γ � Z is a finite set of steps.

CΓn : Γ-paths of length n.

CΓ � �nPN CΓn : Γ-paths.

Lattices of paths

Distributivity

Lattices of paths

Distributivity

Lattices of paths

Distributivity

Lattices of paths

Counterexamples

Notice that there are cases in which we don’t get lattices.

CΓ5 , Γ � t�1, 1, 2u.

Lattices of paths

Counterexamples

Notice that there are cases in which we don’t get lattices.

CΓ5 , Γ � t�1, 1, 2u.

Lattices of paths

Counterexamples

Moreover, even when we get lattices, it may happen that they are notdistributive.CΓ

4 , Γ � t�1, 0, 2u.

Notice that UDDH ^ HUDD � HHHH, which is not the coordinatewisemeet of the two paths.

Lattices of paths

Counterexamples

Moreover, even when we get lattices, it may happen that they are notdistributive.CΓ

4 , Γ � t�1, 0, 2u.

Notice that UDDH ^ HUDD � HHHH, which is not the coordinatewisemeet of the two paths.

Lattices of paths

Main result

Theorem (F., Pinzani (2005))CΓn is a (finite distributive) lattice with respect to coordinatewise meet

and join if and only if

p∆Γ � Γq X rγ�, γ�s � Γ,

� γ� � max Γ, γ� � min Γ;

� ∆Γn � t

°ni�1pxi � yi q | xi , yi P Γ, xi � yj ,@i , ju;

� ∆Γ � �nPN ∆Γn,

Lattices of paths

Main result

� ∆Γn � t

� ∆Γ � �nPN ∆Γn,

Lattices of paths

Main result

� ∆Γn � t

� ∆Γ � �nPN ∆Γn,

Lattices of paths

Main result

� ∆Γn � t

� ∆Γ � �nPN ∆Γn,

Lattices of paths

Some consequences

Corollary (F.,Pinzani (2005))(Motzkin) If Γ is an interval (i.e. Γ � rγ�, γ�s), then CΓ

n is a lattice.

Corollary (F.,Pinzani (2005))(Dyck) If Γ � t�b, au (a, b P N), then CΓ

n is a lattice.

Lattices of paths

Some consequences

Corollary (F.,Pinzani (2005))(Motzkin) If Γ is an interval (i.e. Γ � rγ�, γ�s), then CΓ

n is a lattice.

Corollary (F.,Pinzani (2005))(Dyck) If Γ � t�b, au (a, b P N), then CΓ

n is a lattice.

Distributive lattice structure

Representations

Lattices of Dyck-like pathsA Dyck-like path of type pa, bq is a t�b, au-path. They are a naturalgeneralization of Dyck paths, already considered by Duchon (2000).More recently, see also Bergeron, Preville-Ratelle (2012) and Banderier,Wallner (2015).

Lattices of Dyck-like paths will be denoted Dpa,bqn . Not new: they are

Young lattices in disguise!

q qq q

q qq qq q

qq q q q�

��

AAAAAA

AAAAAAA

AAAAAAAA

AAAAAAAAAA

ppppppp pppppp pppppppppp pppp ppp ppp pp p pÓ

Representations

Lattices of Dyck-like paths

The distributive lattice Dpa,bqn is isomorphic to the dual of the Young

lattice Yλpa,bqn

Proposition (F., Munarini (2011))Let gcdpa, bq � 1. For any n P N,

λpa,bqn � pλn,b, . . . , λn,1, λn�1,b, . . . , λn�1,1, . . . , λ2,b, . . . , λ2,1, λ1,b, . . . , λ1,2q

is a partition with nb � 1 parts, where λh,k � ph � 1qa� tpk � 1q � a{bu.In particular,

|λpa,bqn | � abnpn � 1q

2� n|λpa,bq1 |.

Representations

Join-irreducibles

� Dyck r r r r r r r r r r r r r r r r r��@@��@@��@

@@@��@@��@@��@@

0 2i 2j 2n� Motzkin

q q q q q q q q q q q q q q q q q��@

0 i j n

� Schroder

q q q q q q q q q q q q��@

0 2i 2j 2n

orq q q q q q q q q q q�

� @@

0 2i 2j 2n

Representations

Join-irreducibles

@@@��@@��@@��@@

0 i j n

� Schroder

0 2i 2j 2n

� @@

0 2i 2j 2n

Representations

Join-irreducibles

@@@��@@��@@��@@

0 i j n

� Schroder

0 2i 2j 2n

� @@

0 2i 2j 2n

Representations

Birkhoff representations

Poset of join-irreducibles:

� The spectrum of Dn is isomorphic to the poset of the intervals of achain Cn�2 with n � 1 elements, i.e.SpecpDnq � tpi , jq P C2

n | i ¤ j � 2u � IntpCn�2q.� The spectrum of Mn is isomorphic to the poset of the intervals of

even length of a chain Cn having n � 1 elements, i.e.SpecpMnq � tpi , jq P C2

n | Dk P N such that j � i � 2k � 2qu.� SpecpSnq � tpi , j , kq P C2

n � C1 | j � i ¥ k � 1u.

Representations

n � C1 | j � i ¥ k � 1u.

Representations

n � C1 | j � i ¥ k � 1u.

Representations

qqd qd qdq q qqd q qdq qqqd

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q q qq qq��

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qqb qb qb qbq q q q q qq q q qqb q qbq qq

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q q qq qq qqq��@@��@@

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SpecpS3q

The Euler Characteristic

The Euler characteristic

Given a (finite) distributive lattice, an interesting problem is that ofdetermining its Euler characteristic.

The Euler characteristic is a classical invariant measure which plays animportant role, for instance, in combinatorial geometry and in geometricprobability.

The combinatorial interest of the Euler characteristic lies in its deeprelation with the Mobius function, revealed by G.-C. Rota.

Valuations of distributive lattices

The Euler characteristic of a distributive lattice is a particular valuation.

A valuation on a distributive lattice D with values in R is a functionν : D Ñ R such that νpp0q � 0 and νpx _ yq � νpx ^ yq � νpxq � νpyqfor every x , y P D. A valuation on a finite distributive lattice D isuniquely determined by the values it takes on the set of join-irreduciblesof D, and these values can be arbitrarily assigned.

The (Euler) characteristic of D is defined as the unique valuation χ suchthat χpp0q � 0 and χpxq � 1 for every join-irreducible x of D. Inparticular, χpDq � χpp1q.Is there a nice way of interpreting the Euler characteristic inside ourlattices of paths?

Dyck-like lattices

In the Dyck and Schroder case, the combinatorial interpretation of theEuler characteristic follows from a more general algebraic approach.

A Dyck-like lattice is a distributive lattice whose spectrum is a rankedposet admitting a labelling of its elements with the following properties:all labels are positive integers and every antichain S � ts1, . . . , snu ofjoin-irreducibles can be linearly ordered so that the labels of the elementsof S are distinct and, if s1 and sn are the elements having minimum andmaximum labels, respectively, then s1 ^ sn � s1 ^ s2 ^ � � � ^ sn�1 ^ sn.

Dyck-like lattices

In the Dyck and Schroder case, the combinatorial interpretation of theEuler characteristic follows from a more general algebraic approach.

A Dyck-like lattice is a distributive lattice whose spectrum is a rankedposet admitting a labelling of its elements with the following properties:all labels are positive integers and every antichain S � ts1, . . . , snu ofjoin-irreducibles can be linearly ordered so that the labels of the elementsof S are distinct and, if s1 and sn are the elements having minimum andmaximum labels, respectively, then s1 ^ sn � s1 ^ s2 ^ � � � ^ sn�1 ^ sn.

Dyck-like lattices

Proposition (F., Munarini (2011))For any a, b P N and for every n P N, the lattice Dpa,bq

n is a Dyck-likelattice.

Conjecture (F., Munarini (2011))Every finite Dyck-like lattice can be represented as a sublattice of alattice of Dyck-like paths of suitable length.

Dyck-like lattices

Proposition (F., Munarini (2011))For any a, b P N and for every n P N, the lattice Dpa,bq

n is a Dyck-likelattice.

Conjecture (F., Munarini (2011))Every finite Dyck-like lattice can be represented as a sublattice of alattice of Dyck-like paths of suitable length.

Dyck-like lattices

We will say that an element x of a distributive lattice D isquasi-join-irreducible when there exists an ordered k-tuple ps1, . . . , skqforming an antichain of join-irreducibles such that x � s1 _ � � � _ sk andsi ^ si�1 � p0, for every i � 1, 2, . . . , k � 1.

A special Dyck-like lattice is a Dyck-like lattice where the meet of any

two join-irreducibles is p0 or a join-irreducible. Notice that Dpa,bqn is a

special Dyck-like lattice.

PropositionIn a special Dyck-like lattice D, every quasi-join-irreducible element hasEuler characteristic equal to 1.

Dyck-like lattices

An element x of a finite distributive lattice is said to have aquasi-join-irreducible decomposition when it can be expressed as a join ofquasi-join-irreducible elements x1, . . . , xk such that xi ^ xj � p0, for everyi � j .

PropositionEvery element x � p0 of a finite Dyck-like lattice has an essentially uniquequasi-join-irreducible decomposition.

TheoremLet D be a finite special Dyck-like lattice. Then, for every x P Dztp0u,χpxq is the number of quasi-join-irreducibles in a decomposition of x .

Dyck-like lattices

Dyck and Schroder lattices

In the case of Dyck paths, the characteristic can be interpretedcombinatorially as follows.

TheoremA Dyck path x P Dn is quasi-join-irreducible if and only if it has preciselyone 1-tunnel. Therefore the characteristic of a Dyck path is the numberof its 1-tunnels.

The case of Schroder lattices is completely analogous, since they arespecial Dyck-like lattices. In particular, we have the following result.

TheoremA Schroder path is quasi-join-irreducible if and only if it has exactly one0-tunnel. Therefore the characteristic of a Schroder path equals thenumber of its 0-tunnels.

Dyck and Schroder lattices

In the case of Dyck paths, the characteristic can be interpretedcombinatorially as follows.

TheoremA Dyck path x P Dn is quasi-join-irreducible if and only if it has preciselyone 1-tunnel. Therefore the characteristic of a Dyck path is the numberof its 1-tunnels.

The case of Schroder lattices is completely analogous, since they arespecial Dyck-like lattices. In particular, we have the following result.

TheoremA Schroder path is quasi-join-irreducible if and only if it has exactly one0-tunnel. Therefore the characteristic of a Schroder path equals thenumber of its 0-tunnels.

Motzkin lattices

Motzkin lattices are more difficult, since they are not special. In fact, ifthe meet of two join-irreducibles is � p0, then it is not necessarily ajoin-irreducible.

q q q q q q q q q qq q q q q q q q q�� @

��

Need for an alternative approach...

First of all, we need to understand the meet of two genericjoin-irreducibles.

Motzkin lattices

Motzkin lattices are more difficult, since they are not special. In fact, ifthe meet of two join-irreducibles is � p0, then it is not necessarily ajoin-irreducible.

q q q q q q q q q qq q q q q q q q q�� @

��

Need for an alternative approach...

First of all, we need to understand the meet of two genericjoin-irreducibles.

Motzkin lattices

Truncated pyramid: sequence of k ¥ 1 up steps followed by a sequenceof m ¥ 1 horizontal steps followed by a sequence of k down steps, i.e.UkHmDk (k : dimension, m: length, h: height (if the horizontal steps lieon the line y � h)).

Tn,m,k : set of Motzkin paths of length n having only horizontal steps atheight 0, except for a unique truncated pyramid of dimension k andlength m.

p p p p p p p p p p p p p p p p p�� @

PropositionIn Mn the meet of two join-irreducibles is either p0 or a join-irreducible oran element of Tn,1,k , k ¤ n � 2 (i.e., a Motzkin path with a uniquetruncated pyramid of length 1).

Motzkin lattices

LemmaIf x P Tn,m,h, then χpxq � p�1qh�1m � 1.

PropositionLet x PMn be a quasi-join-irreducible. Then χpxq � opxq � epxq � 1,where opxq is the number of horizontal steps at odd height and epxq isthe number of horizontal steps at even nonzero height (i.e. at evenheight and not lying on the x-axis).

TheoremThe characteristic of a Motzkin path x � p0 is χpxq � }x} � o1pxq � epxq,where }x} denotes the number of quasi-join-irreducibles in adecomposition of x .

Motzkin lattices

PropositionA Motzkin path is quasi-join-irreducible if and only if either it has aunique elevated factor having no horizontal steps at height 1 or it has aunique peak at height 1.

TheoremThe characteristic of a Motzkin path x PMn is

χpxq � opxq � epxq � t1pxq � f1pxq � p1pxq � r1pxq,

where fhpxq, phpxq, thpxq and rhpxq are respectively the number oftruncated pyramids, peaks, maximal sequences of consecutive 1-tunnelsand reverse truncated pyramid of height h in x .

Motzkin lattices

PropositionA Motzkin path is quasi-join-irreducible if and only if either it has aunique elevated factor having no horizontal steps at height 1 or it has aunique peak at height 1.

TheoremThe characteristic of a Motzkin path x PMn is

χpxq � opxq � epxq � t1pxq � f1pxq � p1pxq � r1pxq,

where fhpxq, phpxq, thpxq and rhpxq are respectively the number oftruncated pyramids, peaks, maximal sequences of consecutive 1-tunnelsand reverse truncated pyramid of height h in x .

Unimodality

Statement of the problem

Is the Dyck (resp. Motzkin, Schroder) lattice Dn (resp. Mn, Sn)rank-unimodal, for all n P N?

(A finite sequence a1, a2, . . . an is called unimodal whenever there existsi ¤ n such that a1 ¤ a2 ¤ � � � ¤ ai�1 ¤ ai ¥ ai�1 ¥ � � � an�1 ¥ an).

Equivalent formulations of the same problem:

� Given λ � pn, n � 1, . . . 2, 1q, is the Young lattice Yλ rank-unimodal,for all n P N? (Stanton, 1990).

� Are Dyck paths unimodal with respect to the area? (Bonin, Shapiro,Simion, 1993).

� Is the inv statistic unimodal on 312-avoiding permutations(???...source...???)?

Unimodality

Unfortunately...

We have not been able to solve this long-standing open problem (even ifthere is some strong computational evidence supporting a positiveanswer, also in the Motzkin and Schroder cases).

However, we have proposed a presumably new approach based on theso-called ECO methodology (Barcucci, Del Lungo, Pergola, Pinzani(1999)).

Unimodality

Unfortunately...

We have not been able to solve this long-standing open problem (even ifthere is some strong computational evidence supporting a positiveanswer, also in the Motzkin and Schroder cases).

However, we have proposed a presumably new approach based on theso-called ECO methodology (Barcucci, Del Lungo, Pergola, Pinzani(1999)).

Unimodality

An ECO construction for Dyck paths

Local rule for constructing Dyck paths of semilength n � 1 starting fromDyck paths of semilength n: add a peak in each of the points of the lastdescent.

Unimodality

Why this construction?

This is a nice construction in that it provides a partition of the set ofDyck paths having the same length.

Such a partition of Dn is also interesting from an order-theoretic point ofview. In fact, blocks are saturated chains of Dn. So we have a saturatedchain partition of each Dn.

rr rr rrr rr rr r r r

��

ppppp pppppppppp pppppppppp ppppp ppppp

ppppppppppppppp ppppppppppppppp ppppppppppppppp ppppppppppppppp ppppppppppppppp

Unimodality

We claim that this saturated chain partition plays a relevant role inproving that Dyck lattices are rank-unimodal...

... even if we have not been able to show how yet...!

However, we have found some results concerning the polynomials Ppkqn pxq

which describe the distribution of area and length of the last descent inDyck paths. These polynomials already appeared in works of Carlitz(1972) and subsequently Krattenthaler (1989) (even if from a slightlydifferent perspective).

Unimodality

Main results

� Two recursions (F., 2013):

Ppkqn pxq � xk � pPpk�1q

n�1 pxq � � � � � Ppn�2qn�1 pxqq

Ppkqn pxq � x � pPpk�1q

n pxq � xk�1Ppk�2qn�1 pxqq

� A succession rule which generates the Ppkqn pxq’s (F., 2013):

" p00qpαβqù pα0qppα� 1q1q � � � ppα� βqβqppα� β � 1qβ�1q .

Unimodality

Main results

� Two recursions (F., 2013):

Ppkqn pxq � xk � pPpk�1q

n�1 pxq � � � � � Ppn�2qn�1 pxqq

Ppkqn pxq � x � pPpk�1q

n pxq � xk�1Ppk�2qn�1 pxqq

� A succession rule which generates the Ppkqn pxq’s (F., 2013):

" p00qpαβqù pα0qppα� 1q1q � � � ppα� βqβqppα� β � 1qβ�1q .

Unimodality

Main results

00 11 10 21 32

00 11 10 21 32 10 21 20 31 42 30 41 52 63

For instance, Pp1q5 pxq gives the distribution at level 5-2=3 of the labels

Pp1q5 pxq � x � 2x2 � x3 � x4.

Unimodality

Main results

00 11 10 21 32

00 11 10 21 32 10 21 20 31 42 30 41 52 63

For instance, Pp1q5 pxq gives the distribution at level 5-2=3 of the labels

Pp1q5 pxq � x � 2x2 � x3 � x4.

Heyting algebra structure

Dyck algebras

A Heyting algebra is a lattice H with minimum 0 and maximum 1 suchthat the relative pseudocomplement of x with respect to y exists for allx , y P H. By definition, the relative pseudocomplement of x with respectto y is the element x ù y defined as follows:

x ù y � maxtz P H | x ^ z ¤ yu.

Fact: Every finite distributive lattice is a Heyting algebra (in a canonicalway).

Since Dyck lattices are finite distributive, it could be interesting to exploretheir Heyting algebra structure, as well as its logic-theoretic aspects.

In particular, Heyting algebra properties of Dyck lattices has beeninvestigated also by Muhle (2015).

Dyck algebras

Relative pseudocomplements

Proposition (F. (2015+))Given P,Q P Dn, P ù Q is obtained from Q by replacing thoseportions of path in which P lies weakly below Q with the highest possibleDyck factors.

x0 x1 x2 x3 x4 x5

Figure: P is red, Q is blue and Pù Q is green.

Dyck algebras

Pseudocomplement and regular elements

In a Heyting algebra, the pseudocomplement of x is defined as�x � x ù 0. It can be shown that x ¤��x . The converse, however,does not hold in general. An element x is said to be regular wheneverx ��x . The subposet of regular elements of a Heyting algebra forms aBoolean algebra.

Proposition (F. (2015+))Let P P Dn. Then �P � P ù 0 is obtained from P by

1. replacing each sequence of consecutive hills with a pyramid ofsuitable length and height, and

2. completing the path by suitably adding a (finite) set of hills.

Dyck algebras

Figure: A Dyck path (black) and its pseudocomplement (green).

Dyck algebras

Proposition (F. (2015+))A Dyck path is regular if and only if its factors are all pyramids.

211 121 112

31 22 13

Figure: The Boolean algebra of regular elements of D4 and its isomorphicrepresentation in terms of compositions of 4.

The logic of subintervals

Interval temporal logicIs it possible to give a logic-theoretic interpretation of Dyck algebras?

Yes, using interval temporal logic!

A temporal logic is essentially a kind of logic which allows to deal withstatements whose truth values can vary in time.

An interval temporal logic is characterized by the fact that the truth of astatement depends on the time interval it is evaluated on (rather thanthe time instant).

Typically useful in computer science (when it is important to work withproperties which remain true or false for a certain amount of time):

� processes, in particular concurrent real-time processes,

� temporal databases,

� specification, design and verification of hardware components.

Interval temporal logic

There is a large literature on interval temporal logics. In particular,fragments of specific logics are investigated from the point of view ofcomputability and computational complexity (Goranko, Montanari,Sciavicco (2004), Montanari, Pratt-Hartmann, Sala (2010), Bresolin,Della Monica, Montanari, Sala, Sciavicco (2014) ...)

We will focus on a specific fragment of the whole Halpern-Shoham logic,which is sometimes called the logic of subintervals. In a sense, this is anintermediate step between genuine interval logics and classical logics (oftime instants rather than intervals). So, it is probably not among themost expressive fragments from a purely logic-theoretic point of view (itsdecidability has been studied by Marcinkowski and Michaliszyn (2014)).However, it has a very nice and effective combinatorial description, whichwe are going to develop in the next slides.

Interval temporal logic

There is a large literature on interval temporal logics. In particular,fragments of specific logics are investigated from the point of view ofcomputability and computational complexity (Goranko, Montanari,Sciavicco (2004), Montanari, Pratt-Hartmann, Sala (2010), Bresolin,Della Monica, Montanari, Sala, Sciavicco (2014) ...)

We will focus on a specific fragment of the whole Halpern-Shoham logic,which is sometimes called the logic of subintervals. In a sense, this is anintermediate step between genuine interval logics and classical logics (oftime instants rather than intervals). So, it is probably not among themost expressive fragments from a purely logic-theoretic point of view (itsdecidability has been studied by Marcinkowski and Michaliszyn (2014)).However, it has a very nice and effective combinatorial description, whichwe are going to develop in the next slides.

The general framework

� Tn � tt1, t2, . . . , tnu: finite linearly ordered set of time states.

� IntpTnq: set of intervals of Tn.

� IntpTnq is partially ordered by inclusion.

Set of propositions ITLn recursively defined as follows:

� K,J P ITLn; for all 1 ¤ i ¤ n, εi P ITLn (propositional variables);

� if ϕ,ψ P ITLn, then ϕ_ ψ,ϕ^ ψ,ϕÑ ψ, ϕ P ITLn.

Interval-based semantics:

v : ITLn ÝÑ 2IntpTnq

: ϕ ÞÝÑ vϕ : IntpTnq ÝÑ t0, 1u

where vϕpI q � ϕpI q � 0 (resp., 1) if ϕ is false (resp., true) whenevaluated on the interval I .

We consider a specific evaluation v :

� εi pI q � 1 whenever I � ttiu;� pϕ_ ψqpI q � 1 whenever ϕpI q � 1 or ψpI q � 1;

� pϕ^ ψqpI q � 1 whenever ϕpI q � 1 and ψpI q � 1;

� p ϕqpI q � 1 whenever ϕpI q � 0;

� pϕÑ ψqpI q � 1 whenever if ϕpI q � 1 then ψpI q � 1.

Two new connectives:

� plϕqpI q � 1 when, for all intervals J � I , ϕpJq � 1;

� p♦ϕqpI q � 1 when there exists an interval J � I such that ϕpJq � 1.

ϕ is valid when ϕpI q � 1 for all I P IntpTnq.

Θn � tϕ P ITLn | ϕÑ lϕ is validu.

Translation: if ϕ is true in I , then ϕ is true in all subintervals of I .

Behavior of Θn with respect to classical connectives:

Proposition (F. (2015+))

� ϕ,ψ P Θn ñ ϕ^ ψ,ϕ_ ψ P Θn.

� Dϕ P Θn : ϕ R Θn.

� Dϕ,ψ P Θn : ϕÑ ψ R Θn.

We need better negation and implication!

Define two new connectives � (pseudonegation) and ù(pseudoimplication) with the following semantics:

� p�ϕqpI q � 1 whenever @J � I , ϕpJq � 0;

� pϕù ψqpI q � 1 whenever @J � I , if ϕpJq � 1, then ψpJq � 1.

ϕ,ψ P Θn ñ�ϕ, ϕ ù ψ P Θn.

Pseudonegation has the typical behavior of an intuitionistic negation.

� ϕpI q � 1 ñ p��ϕqpI q � 1, but the converse does not hold ingeneral;

� p�ϕqpI q � 1 ô p��ϕqpI q � 1.

Pseudonegation has the typical behavior of an intuitionistic negation.

� ϕpI q � 1 ñ p��ϕqpI q � 1, but the converse does not hold ingeneral;

� p�ϕqpI q � 1 ô p��ϕqpI q � 1.

The isomorphism theorem

� ϕ () ψ when vpϕq � vpψq.� () is an equivalence relation on Θn which preserves _,^,ù,�.

� Canonical distributive lattice structure on the quotient Θn{().

Proposition (F. (2015+))For every ϕ,ψ P Θn, we have:

rϕsù rψs �ªtrαs P Θn{() | rϕs ^ rαs ¤ rψsu.

In other words, ù is the relative pseudocomplement operation in thecanonical Heyting algebra structure of rΘns.

Lemma (F. (2015+))For any ϕ P ITLn, set ϕ ��ϕ. Given an interval I of rns, set εI �

�iPI εi . Then,

for any ϕ P Θn, there exists an antichain of intervals I1, I2, . . . , Ir of rns such that

ϕ () εI1 _ εI2 _ � � � _ εIr .

Moreover, when the intervals are listed in increasing order of their minima, the aboveone is the unique proposition of that form equivalent to ϕ.

This will be called the closed disjunctive form (CDF) of ϕ.

Theorem (F. (2015+))The Heyting algebra Dn of Dyck paths of semilength n is isomorphic tothe Heyting algebra rΘn�1s.Therefore, a Dyck path of semilength n can be identified with a suitableantichain of intervals of rn � 1s.

�iPI εi . Then,

ϕ () εI1 _ εI2 _ � � � _ εIr .

�iPI εi . Then,

ϕ () εI1 _ εI2 _ � � � _ εIr .

�iPI εi . Then,

ϕ () εI1 _ εI2 _ � � � _ εIr .

Posets of intervals

A more general point of view:

� Given a finite poset P, consider IntpPq (intervals of P) ordered bycontainment.

� Study the Heyting algebra OpIntpPqq obtained via Birkhoff theorem(P is the set of atoms of OpIntpPqq).

� Interesting cases (|P| � n):

1. P discrete ñ OpIntpPqq � Bn (Boolean algebra having n atoms);2. P linearly ordered ñ OpIntpPqq � Dn�1;3. P Boolean algebra ñ IntpPq � cubical lattices of faces of an n-cubeñ OpIntpPqq �??? (Rota, Metropolis (1978), Bailey, Oliveira(1998), Mundici (2016)).

Posets of intervals

Combinatorial statistics on Dyck paths

Is it possible to recover combinatorial properties of Dyck paths from theHeyting algebra structure of Dn and from the linear order structure of itsatoms?

P a Dyck path of semilength n.

� FP � tI1, . . . , Imu: antichain of intervals of rn � 1s associated withP;

� |FP | � m: number of intervals of FP ;

� }FP} � |I1 Y � � � Y Im|.

Posets of intervals

� }FP} � |I1 Y � � � Y Im|.

Posets of intervals

� }FP} � |I1 Y � � � Y Im|.

Posets of intervals

� }FP} � |I1 Y � � � Y Im|.

Posets of intervals

A bunch of interesting statistics (F. (2015+)):

� Number of peaks of P: |FP | � }F�P} � |F��P |.

� Number of hills of P: }F�P} � |F��P |.

� Sum of the heights of the peaks of P:}FP} � |FP | � }F�P} � |F�