A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths

Journal of Mathematical Psychology 54 (2010) 471–474

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A bijection between a set of lexicographic semiorders and pairs of non-crossing Dyck paths Clintin P. Davis-Stober University of Missouri, United States

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Article history: Received 1 June 2009 Received in revised form 1 July 2010 Available online 25 September 2010 Keywords: Dyck paths Non-crossing Dyck paths Semiorders Lexicographic semiorders Catalan numbers

abstract It is a known result that the set of distinct semiorders on n elements, up to permutation, is in bijective correspondence with the set of all Dyck paths of length 2n. I generalize this result by defining a bijection between a set of lexicographic semiorders, termed simple lexicographic semiorders, and the set of all pairs of non-crossing Dyck paths of length 2n. Simple lexicographic semiorders have been used by behavioral scientists to model intransitivity of preference (e.g., Tversky, 1969). In addition to the enumeration of this set of lexicographic semiorders, I discuss applications of this bijection to decision theory and probabilistic choice. © 2010 Elsevier Inc. All rights reserved.

1. Introduction A lexicographic semiorder is a generalization of a semiorder primarily used by decision theorists to develop and test models of intransitive preference (Birnbaum, 2010; Birnbaum & Gutierrez, 2007; Grether & Plott, 1979; Roelofsma & Read, 2000; Tversky, 1969). The key feature of a decision-making model based on a lexicographic semiorder is that a decision maker examines the individual attributes (features) of choice alternatives sequentially, preferring one over another if, and only if, the difference between the attributes exceeds some pre-determined threshold (i.e., a semiorder). As an example, suppose that a decision maker is deciding between the lotteries a = ($100; 0.33) and b = ($95; 0.40).1 The decision maker may first examine the payoffs of the lotteries and decide that the difference between $100 and $95 is not sufficiently great. The decision maker may then examine the probabilities of winning and decide that 0.40 is sufficiently larger than 0.33 and prefer lottery b to a. Let A be a non-empty set such that |A| = n. A binary relation, S, is a (strict) semiorder if, and only if, there exists a real-valued function g, defined on A, and a non-negative constant q such that, ∀a, b ∈ A, aSb ⇐⇒ g (a) > g (b) + q.

(1)

Note that the relation S is asymmetric. Semiorders were first developed by Luce (1956) to model intransitive indifference in decision-making. Wine and Freund (1957) demonstrated that the number of nonisomorphic over n elements is the  semiorders  2n

1 nth Catalan number, n+ ,—see also Chandon, Lemaire, and 1 n Pouget (1978). Define the trace of a semiorder as the relation T such that

aTb ⇐⇒ [bSc ⇒ aSc , ∀c ∈ A]

M ki ≤ M kj ,

∀(i, j, k) ∈ {1, 2, . . . , n} : i ≤ j,

(2)

ik

≥M ,

∀(i, j, k) ∈ {1, 2, . . . , n} : i ≤ j,

(3)

i,i

= 0,

M 1 A lottery x = ($X ; p) is read as a first-order binary lottery where a dollar amount $X is won with probability p, $0 dollars won with probability 1 − p. 0022-2496/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2010.09.001

[dSa ⇒ dSb, ∀d ∈ A].

In general, T defines a weak ordering on A. A linear ordering of A is compatible with T if it is contained in T , i.e., if T ∗ is a compatible linear order for T , then for all a, b ∈ A, aT ∗ b ⇒ aTb. Semiorders can be equivalently characterized in terms of a strictly upper triangular step matrix (Mirkin, 1972; Sharp, 1972). Let S be a semiorder with trace T . Let M be a matrix of size n × n with rows and columns indexed by the elements of A with indices descending according to T . Let M ij = 1 ⇐⇒ iSj with M ij = 0 otherwise. This definition uniquely associates to each distinct semiorder a ‘step-style’ matrix, meaning that if an entry is equal to 1 then so are all entries to the right and above. The set of all such semiorder matrices is the set of all 0/1 matrices of size n × n that satisfy the following system of linear (in)equalities:

M E-mail address: [email protected].

and

jk

∀i, i ∈ {1, 2, . . . , n}.

(4)

A lexicographic semiorder is a binary relation that can be characterized by an ordered collection of semiorders (—see,

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C.P. Davis-Stober / Journal of Mathematical Psychology 54 (2010) 471–474

e.g., Pirlot & Vincke, 1997). Define a lexicographic semiorder as the following relation P L on A such that there exists a collection of kmany semiorders S1 , . . . , Sk on A, such that, ∀a, b, ∈ A aP L b ⇐⇒ ∃j : aSj b

and ¬[aSi b or bSi a],

∀i < j.

In this article, I consider a special case of a lexicographic semiorder that is characterized by two semiorders and is defined as follows. Definition. A binary relation P is a simple lexicographic semiorder if, and only if, there exist two semiorders S1 and S2 on A, such that, ∀a, b ∈ A:

Fig. 1. This figure displays the Dyck path f = uuddudud, which is of length 8 (n = 4).

• the trace of S1 is the reversal of the trace of S2 ; • aPb ⇐⇒ aS1 b, or [aS2 b and ¬[aS1 b or bS1 a]]. The trace of a semiorder S1 is the reversal of the trace of S2 if, and only if, there exists a linear ordering compatible with both the weak order defined by the trace of S1 and the reverse of the weak order defined by the trace of S2 . Note that P is asymmetric and not necessarily transitive. Variations of the relation P have been used by behavioral scientists to model intransitivity of preference (e.g. Tversky, 1969). See Pirlot and Vincke (1992, 1997) for alternative definitions of lexicographic semiorders that preserve transitivity. In this article, I present a bijection between the set of all distinct simple lexicographic semiorders and the set of all pairs of non-crossing Dyck paths of length 2n. Dyck paths and pairs of non-crossing Dyck paths are well-studied mathematical structures with important applications in combinatorics, coding theory, and statistics. This bijection allows us to enumerate the cardinality of the set of simple lexicographic semiorders as a function of the Catalan numbers. I also discuss applications of this bijection to decision theory. 2. Dyck paths and non-crossing Dyck paths A Dyck path of length 2n is defined as a path in the grid

{0, 1, 2, . . . , 2n}2 that begins at (0, 0) and ends at the point (2n, 0) and never goes below the x-axis. Dyck paths are comprised of two basic movements. Define the Dyck alphabet D = {u, d}, where ‘u’ is a step of +1 along both axes of the grid {0, 1, 2, . . . , 2n}2 (i.e., a North-East step), and ‘d’ is a step of +1 along the x-axis and −1 along the y-axis in the grid {0, 1, 2, . . . , 2n}2 (a South-East step). Let a word be a concatenation of letters from D. A word f ′ is a prefix of f if there exists a word f ′′ such that f = f ′ f ′′ . Let Dn be defined as the set of all words of length 2n that conform to the definition of a Dyck path. Definition. Let δ : Dn → Z, be a mapping from Dn to the integers defined as follows

δ(f ) = δ(a1 a2 · · · a2n ) =

2n −

Fig. 2. This figure shows a pair of non-crossing Dyck paths, g = uuuudddd and f = uuddudud. Both Dyck paths are of length 8.

between a class of Motzkin paths (a generalization of Dyck paths) and a class of semiorders. A pair of Dyck paths (f , g ) is non-crossing if, and only if, they are of equal length and δ(g ′ ) ≥ δ(f ′ ), for all prefixes f ′ , g ′ of (f , g ). Geometrically, the Dyck path g never dips below the Dyck path f in the grid {1, 2, . . . , 2n}2 . Let Dn∗ be defined as the set of all ordered pairs of non-crossing Dyck paths of length 2n. Fig. 2 shows a pair of non-crossing Dyck paths, g = uuuudddd and f = uuddudud. Sets of pairs of non-crossing Dyck paths are in bijective correspondence with many structures. Elizalde (2007) defines a bijection between 2-triangulations of a convex n-gon and pairs of non-crossing Dyck paths of length 2(n − 4). Bonichon (2005) defines a bijection between pairs of non-crossing Dyck paths and realizers of maximal plane graphs, where realizers are defined as a decomposition of a plane graph into Schnyder trees. 3. Semiorders and Dyck paths To see that the set of all distinct semiorders over n alternatives is in bijective correspondence with the set of all Dyck paths of length 2n, we define the following mapping. Let f ∈ Dn . First, we number the ‘d’ elements of f consecutively from 1 to n beginning with the first ‘d’ element in the Dyck word. As an example, the Dyck path f = uudududd, is of length 8 and would be numbered

δ(ai ),

1

where ∀i, ai ∈ D, and δ(u) = 1 and δ(d) = −1. Note that ∀f ∈ Dn , δ(f ) = 0 and for all prefixes of f , denoted f ′ , δ(f ′ ) ≥ 0. Fig. 1 displays the Dyck path f = uuddudud, which is of length 8 (n = 4). Similar to semiorders, it is well-known that the number of distinct Dyck paths of length 2n is the nth Catalan number (e.g., Stanley, 1999), see also Deutsch (1999). Stanley (1999) gives an overview of the bijective relationships between Dyck paths and many other mathematical structures, including binary trees (see also Gu, Li, & Mansour, 2008), triangulations of an (n + 2)-gon, ways to parenthesize a string of length n + 1, and ballot sequences. Balof and Menashe (2007) present a bijection

f = uud1 ud2 ud3 d4 . f

Let pdi q be the number of u letters preceding di for a Dyck word f

f

f

f ∈ Dn . For the above example, pd1 q = 2, pd2 q = 3, pd3 q = 4, and

f pd4 q

= 4.

Definition. Let µ : Dn → {0, 1}n×n , f → µ(f ) = Mf , be a mapping from the set of all Dyck paths of length 2n to the set of 0/1 matrices that satisfy inequalities (2)–(4), defined as follows ij

Mf :=



f

1 ⇔ pdi q < j, 0 otherwise. f

This definition gives a matrix, Mf , in which row i begins with pdi q many 0’s with the remainder of the row entries equal to 1. Fig. 3

C.P. Davis-Stober / Journal of Mathematical Psychology 54 (2010) 471–474

Fig. 3. This figure illustrates the mapping µ for a Dyck path of length eight. The ‘step lines’ are drawn on the matrix to help demonstrate the relationship between this Dyck path and its corresponding semiorder matrix.

illustrates the mapping µ for a Dyck path of length eight. The ‘step lines’ are drawn on the matrix to help demonstrate the relationship between the Dyck path of length 2n and its corresponding semiorder matrix. It is easy to see that µ is a bijection; simply rotate the graph of the Dyck path 45°. Proposition 1. Let µ be defined as above. Let (f , g ) be an ordered pair of Dyck paths of length 2n. Then (f , g ) are non-crossing Dyck paths if, and only if, Mf and Mg each satisfy inequalities (2)–(4) and ij

ij

Mf ≥ Mg , ∀(i, j) ∈ {1, 2, . . . , n}. Proof. (⇒) Let (f , g ) be a pair of non-crossing Dyck paths. By definition, Mf and Mg each satisfy the inequalities (2)–(4). Assume

∃(i, j) such that Mfij < Mgij . This implies that pdif q > pdgi q, which implies that there exist prefixes of (f , g ), denoted f ′ and g ′ respectively, such that δ(f ′ ) > δ(g ′ ). This is a contradiction, as (f , g ) were assumed to be non-crossing. The other direction (⇐) is trivial.



4. Lexicographic semiorders and pairs of non-crossing Dyck paths Let P be a simple lexicographic semiorder defined on the set A such that |A| = n. Similar to semiorders, we represent each distinct simple lexicographic semiorder with a binary matrix of size n × n denoted MP . Assume that the rows and columns of MP are indexed by elements of A with indices descending according to the ij trace of S1 and the reverse of the trace of S2 . Let MP = 1 ⇐⇒ iPj ij

with MP = 0 otherwise. The set of all such simple lexicographic semiorder matrices is the set of all 0/1 matrices of size n × n that satisfy the following six linear (in)equalities M ki ≤ M kj ,

∀(i, j, k) ∈ {1, 2, . . . , n} : i ≤ j, k ≤ i, k ≤ j,

(5)

M ik ≥ M jk ,

∀(i, j, k) ∈ {1, 2, . . . , n} : i ≤ j, i ≤ k, j ≤ k,

(6)

M ki + M ik ≤ M kj + M jk ,

∀(i, j, k) ∈ {1, 2, . . . , n} : i ≤ j,

(7)

≥M +M ,

∀(i, j, k) ∈ {1, 2, . . . , n} : i ≤ j,

(8)

ik

M +M M M

ki

1n

+M

i ,i

= 0,

n1

jk

kj

≤ 1, ∀i, i ∈ {1, 2, . . . , n}.

(9) (10)

Similar to the semiorder representation, the inequalities (5)–(6) imply that the upper triangular portion of MP , denoted UP , is a ‘step-style’ matrix that conforms to the inequalities (2)–(4) and represents the semiorder S1 . Let M T denote the transpose of a matrix M . The inequalities (7)–(8) imply that the lower triangular portion of MP , denoted LP , has the property that the matrix K , defined by K = UP + LPT , must also satisfy the semiorder inequalities (2)–(4) and represents the semiorder S2 .

473

Fig. 4. This figure illustrates the mapping γ for a pair of Dyck paths. The two ‘step lines’ are drawn on the lexicographic semiorder matrix to illustrate its relationship with the Dyck path pair.

We now define a bijective mapping between the set of all distinct simple lexicographic semiorders over n elements and the set of all pairs of non-crossing Dyck paths of length 2n. Definition. Let γ : Dn∗ → {0, 1}n×n , (f , g ) → γ (f , g ) = Mfg , denote a mapping from the set of non-crossing Dyck paths of size 2n to the set of 0/1 matrices of size n × n, defined as follows

γ (f , g ) = µ(g ) + µ(f )T − (µ(g ) ◦ µ(f ))T = Mg + MfT − (Mg ◦ Mf )T = Mfg ,

(11)

where ‘‘◦’’ is defined as the Hadamard product, i.e., element-wise multiplication between two matrices of equal size. The mapping γ is well defined, as each matrix is square and of the same size. Furthermore, each Dyck path must have exactly n many ‘u’ elements and exactly n many ‘d’ elements. It is a routine exercise to verify that Mfg satisfies inequalities (5)–(10). See Fig. 4 for an example of this mapping. Lemma 1. Let µ and γ be defined as above. Then γ is an injection from Dn∗ to the set of all 0/1 matrices of size n × n that satisfy (in)equalities (5)–(10). Proof. Let (f , g ), (x, y) ∈ Dn∗ . Assume γ (f , g ) = γ (x, y). This gives Mfg = Mxy , which implies

µ(g ) + µ(f )T − (µ(g ) ◦ µ(f ))T = µ(y) + µ(x)T − (µ(y) ◦ µ(x))T .

(12)

Note that on the left side of (12) only the Dyck path g contributes to the upper triangular portion of Mfg , likewise only the Dyck path y contributes to the upper triangular portion of Mxy . Since µ is a bijection we have Mfg = Mxy ⇒ µ(g ) = µ(y), thus g = y. Substituting g for y in (12) gives, Mg + MfT − (Mg ◦ Mf )T = Mg + MxT − (Mg ◦ Mx )T ,

(13)

⇔Mf − Mx = Mg ◦ (Mf − Mx ).

(14)

If we show that Mf = Mx we are done. Suppose Mf ̸= Mx . Then

∃(i, j) such that Mfij = 0 and Mxij = 1 (or Mfij = 1 and Mxij = ij 0, following an identical argument). This implies that Mg = 1, otherwise Mg ◦ (Mf − Mx ) ̸= Mf − Mx . But this is a contradiction, ij ij as if Mg = 1 and Mf = 0 then Proposition 1 is violated. Therefore Mf = Mx , and since µ is a bijection, f = x. Hence, γ is an injection.



Lemma 2. Let µ and γ be defined as above. Then γ is a surjection from Dn∗ to the set of 0/1 matrices of size n × n that satisfy (in)equalities (5)–(10). Proof. Let MP be a matrix of size n × n that satisfies the inequalities (5)–(10). Let UP and LP be respectively the n × n upper and lower triangular matrices such that UP + LP = MP . The upper triangular matrix UP satisfies the inequalities (2)–(4) by definition and therefore corresponds uniquely to a Dyck path, g0 , through

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the mapping µ. Consider the matrix K = UP + LPT . This matrix also satisfies the inequalities (2)–(4) and hence corresponds to a Dyck path, f0 , through the mapping µ. Clearly K ij ≥ U ij , ∀(i, j) ∈ {1, 2, . . . , n}; hence, by Proposition 1, (f0 , g0 ) are non-crossing Dyck paths giving γ (f0 , g0 ) = MP .  By Lemmas 1 and 2, γ is both an injection and a surjection, therefore γ is a bijection. This result allows us to enumerate the number of distinct simple lexicographic semiorders on n elements.  2n

1 Let Cn be the nth Catalan number, n+ . Gouyou-Beauchamps 1 n (1986) proved that the set of all pairs of non-crossing Dyck paths of length 2n has a cardinality equal to Cn Cn+2 − Cn2+1 . Lemmas 1 and 2 show that the set of all distinct simple lexicographic semiorders also has this cardinality up to isomorphism.

5. Discussion In this article, I defined a bijection between the set of simple lexicographic semiorders and the set of all pairs of non-crossing Dyck paths of length 2n. This result is useful for investigators interested in studying probabilistic decision theory. A key feature of a simple lexicographic semiorder is that the trace of S2 is the reversal of S1 . This property can be used to model preferential choice over alternatives with attributes that ‘‘trade-off’ with one another. For example, ordering a collection of consumer products by their price is often the reverse ordering of those same products by their quality (or features, etc.). DavisStober (unpublished manuscript) developed a model of decisionmaking that models individual choice as a sampling process over simple lexicographic semiorders. Applied to risky choice, this model considers gambles where payoff values trade-off with the probability of winning. The enumerative results in this article provide an explicit characterization of the possible preference states within this model and allow an accurate estimation of the model’s complexity. A variation of this type of lexicographic semiorder ‘mixture model’ was also investigated by Birnbaum and LaCroix (2008). This bijection also allows us to examine individual preference using the pairs of non-crossing Dyck paths. Assume that a decision maker’s preferences conform to a simple lexicographic semiorder. Then the height of the ‘peaks’ in each corresponding non-crossing Dyck path represents which elements of A are easily dissociated for the decision maker. Furthermore, the differences between the corresponding non-crossing Dyck paths at each point reflect how the decision maker differentially evaluates the two attributes that S1 and S2 represent, indexed by the elements of A. As an example, the Dyck path g = uu · · · udd · · · d would correspond to complete indifference (no preferences between elements of A) along the attribute of interest. Also, two identical Dyck paths indicate that a decision maker evaluates each attribute in a similar way. The bijection presented here gives an enumeration of all distinct simple lexicographic semiorders. To my knowledge, the more general problem of enumerating the set of all distinct lexicographic semiorders characterized with k-many semiorders (i.e., S1 , . . . , Sk ) each with arbitrary traces remains an interesting open problem.

Acknowledgments Much of this work was made possible by a pre-doctoral trainee fellowship awarded to the author from the National Institutes of Mental Health under Training Grant Award Nr. PHS 2 T32 MH014257 entitled ‘‘Quantitative Methods for Behavioral Research’’ (to M. Regenwetter, PI) as well as a Dissertation Completion Fellowship awarded to the author by the University of Illinois at Urbana–Champaign. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author and do not necessarily reflect the views of the National Institutes of Mental Health, the University of Illinois at Urbana–Champaign, or the University of Missouri. I gratefully acknowledge Robert Fossum, William Messner, Stephen Broomell, and Michel Regenwetter for their feedback on early drafts of this manuscript. I would also like to thank the action editor, Jean-Paul Doignon, and two anonymous referees for their helpful comments and suggestions. Any inconsistencies or errors are the sole responsibility of the author. References Balof, B., & Menashe, J. (2007). Semiorders and Riordan numbers. Journal of Integer Sequences, 10, Article 07.7.6. Birnbaum, M. H. (2010). Testing lexicographic semiorders as models of decision making: priority dominance, integration, interaction, and transitivity. Journal of Mathematical Psychology, 54, 363–386. Birnbaum, M. H., & Gutierrez, R. J. (2007). Testing for intransitivity of preferences predicted by a lexicographic semiorder. Organizational Behavior and Human Decision Processes, 104, 96–112. Birnbaum, M. H., & LaCroix, A. R. (2008). Dimension integration: testing models without trade-offs. Organizational Behavior and Human Decision Processes, 105, 122–133. Bonichon, N. (2005). A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths. Discrete Mathematics, 298, 104–114. Chandon, J. L., Lemaire, J., & Pouget, J. (1978). Dénombrement des quasi-ordres sur un ensemble fini. Mathématiques et Sciences Humaines, 62, 61–80. Davis-Stober, C. P. (2010). A new perspective on lexicographic semiorder models. University of Missouri at Columbia (unpublished manuscript). Deutsch, E. (1999). Dyck path enumeration. Discrete Mathematics, 204, 167–202. Elizalde, S. (2007). A bijection between 2-triangulations and pairs of non-crossing Dyck paths. Journal of Combinatorial Theory, Series A, 114, 1481–1503. Gouyou-Beauchamps, D. (1986). Chemins sous-diagonaux et tableau de Young. In Lecture notes in mathematics: Vol. 1234. Combinatoire énumérative, Montreal, 1985 (pp. 112–125). Grether, D. M., & Plott, C. R. (1979). Economic theory of choice and the preference reversal phenomenon. The American Economic Review, 69, 623–638. Gu, N. S. S., Li, N. Y., & Mansour, T. (2008). 2-binary trees: bijections and related issues. Discrete Mathematics, 7, 1209–1221. Luce, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24, 178–191. Mirkin, B. G. (1972). Description of some relations on the set of real-line intervals. Journal of Mathematical Psychology, 9, 243–252. Pirlot, M., & Vincke, Ph. (1992). Lexicographic aggregation of semiorders. Journal of Multi-Criteria Decision Analysis, 1, 47–58. Pirlot, M., & Vincke, Ph. (1997). Semiorders: properties, representations, applications. Netherlands: Kluwer Academic Publishers. Roelofsma, P. H. M. P., & Read, D. (2000). Intransitive intertemporal choice. Journal of Behavioral Decision Making, 13, 161–177. Sharp, H. (1972). Enumeration of transitive, step-type relations. Acta Mathematica Academiae Scientiarum Hungaricae, 22, 365–371. Stanley, R. P. (1999). Enumerative combinatorics: Vol. 2. USA: Cambridge University Press. Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76, 31–48. Wine, R. L., & Freund, J. E. (1957). On the enumeration of decision patterns involving n means. The Annals of Mathematical Statistics, 28, 256–259.