Local height in weighted Dyck models of random walks and the variability of the number of coalescent histories for caterpillarshaped gene trees and species trees
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Abstract
We examine combinatorial parameters of three models of random lattice walks with up and down steps. In particular, we study the height \(y_i\) measured after i upsteps in a random weighted Dyck path of size (semilength) n. For a fixed integer \(w \in \{0,1,2\}\), the considered weighting scheme assigns to each Dyck path of size n a weight \(\prod _{i=1}^n y_i^w\) that depends on the height of the upsteps of the path. We investigate the expected value \({\text{E}}_n(y_i)\) of the height \(y_i\) in a random weighted Dyck path of size n, providing exact formulas for \({\text{E}}_n(y_i)\) and \({\text{E}}_n(y_i^2)\) when \(w=0,1\), and estimates of the mean of \(y_i\) for \(w=2\). Denoting by \(i^*(n)\) the position i where \({\text{E}}_n(y_i)\) reaches its maximum \({\text{m}}(n)\), our calculations indicate that, when n becomes large, the pair \(\big (i^*(n), {\text{m}}(n)\big )\) grows like \(\big ( n/2 , 2\sqrt{n/\pi } \big )\) if \(w=0\), \(\big ( 3n/4 , n/2 \big )\) if \(w=1\), and \(\big ( (9+\sqrt{17})n/16 , (1+\sqrt{17})n/8 \big )\) if \(w=2\). These results also contribute to the study of the variability of the number of “coalescent histories”: structures used in models of gene tree evolution to encode the combinatorially different configurations of a gene tree topology along the branches of a species tree. Relationships with other combinatorial and algebraic structures, such as alternating permutations and Meixner polynomials, are also discussed.
Keywords
Lattice walks Weighted Dyck paths Combinatorial enumeration Computational biologyMathematics Subject Classification
05A15 05A16 92B101 Introduction
Lattice walks (or paths) often serve as models of statistical mechanical systems whose physical properties are linked with the combinatorial and enumerative features of the paths under consideration [17]. In this article, we study from a combinatorial point of view models of lattice walks derived from Dyck paths. A Dyck path is a trajectory of a random walk with steps \(U=(+1,+1)\) and \(D=(+1,1)\) on the twodimensional square lattice starting at the origin (0, 0) and ending on the xaxis, where each point on the trajectory has a nonnegative ordinate (Fig. 1a). Several physical systems can be described by means of Dyck paths. For instance, Dyck paths can be seen as wave functions of spins [25] or, under suitable weighting schemes, they are used as polymer models [3]. In combinatorics, enumerative features of Dyck paths have been investigated with respect to several parameters (see e.g. [8]), starting from the number of Dyck paths of given length that is counted by the ubiquitous [9, 27] sequence of Catalan numbers.
Motivated also by computational problems in models of gene tree evolution [7, 10, 11, 23, 24], our paper contributes to the combinatorial analysis of Dyck models of lattice walks by studying the ordinate, or height, \(y_i=y(U_i)\) of the ith up step \(U_i\) in Dyck paths of given size (semilength) taken under three different coloring or, equivalently, weighting schemes. In addition to the set \({\mathcal {D}}_n\) consisting of Dyck paths of size n, we consider the sets of paths \({\mathcal {D}}_n^{\prime }\) and \({\mathcal {D}}_n^{\prime \prime }\). These are obtained by coloring, in all possible ways, the steps of each \(\gamma \in {\mathcal {D}}_n\) as follows. In \({\mathcal {D}}_n^{\prime }\), each up step \((U_i)_{1\le i\le n}\) of \(\gamma\) is colored by an integer label in the range \([1,y(U_i)]\) (Fig. 1b). In \({\mathcal {D}}_n^{\prime \prime }\), also down steps \((D_i)_{1\le i\le n}\) of \(\gamma\) receive colors, with each \(D_i\) colored by an integer label in \([1,y(D_i)]\), where \(y(D_i)\) is the height of \(D_i\) (Fig 1c). Counting paths of \({\mathcal {D}}_n^{\prime }\) and \({\mathcal {D}}_n^{\prime \prime }\) is thus equivalent to counting Dyck paths of size n in which the weight (multiplicity) of each path is given by the product \(\prod _{i=1}^n y_i\) and \(\prod _{i=1}^n y_i^2\), respectively. In particular, the cardinality of \({\mathcal {D}}_n^{\prime }\) is \({\mathcal {D}}_n^{\prime }=(2n1)!!\), whereas \({\mathcal {D}}_n^{\prime \prime }\) is the nth Euler/secant number.
The weighting schemes described above have been already considered in the literature (e.g. [5, 12, 15, 22]) due, for instance, to their interesting relationships with continued fractions. Continued fractions enable the study of the height—the ordinate of the highest point—of a uniform random path of \({\mathcal {D}}_n,{\mathcal {D}}_n^{\prime }\), and \({\mathcal {D}}_n^{\prime \prime }\). For example (see e.g. [13] Section V.4.3), the expected height of a uniform random path of \({\mathcal {D}}_n\) (resp. \({\mathcal {D}}_n^{\prime }\)) is known to grow asymptotically like \(\sqrt{\pi n}\) (resp. n/2). Here, we investigate a slightly different type of statistic: we study, for each fixed position \(i \in [1,n]\), the height \(y_i\) of a random Dyck path of size n, when this is taken with a probability induced by a uniform distribution over the sets \({\mathcal {D}}_n,{\mathcal {D}}_n^{\prime }\), and \({\mathcal {D}}_n^{\prime \prime }\). In each of these three cases, we focus on the value \(i^*(n)\) of i where the expectation of \(y_i\) has its maximum, on the value \({\text{m}}(n)\) of this maximum, and on the mean of \(y_n\)—the expected length of the last descent. The lack of a simple recursive decomposition for paths in \({\mathcal {D}}_n^{\prime \prime }\) makes the study of the height variables \(y_i\) quite difficult in this case. In addition to exact results, we also present empirical estimates based on intuitive, although not completely rigorous, theoretical arguments coupled with numerical validations.
After some preliminary results presented in Sect. 2, Sect. 3 is dedicated to the combinatorial and numerical analysis of the height variables \(y_i\) under the three different lattice models \({\mathcal {D}}_n,{\mathcal {D}}_n^{\prime },\) and \({\mathcal {D}}_n^{\prime \prime }\). Our findings are then shown in Sect. 4 to have interesting relationships with the variability of the number of evolutionary configurations, also called coalescent histories [7, 10, 11, 23, 24], of a gene tree, representing the genetic history of gene copies sampled from individuals, in a species tree, which represents the ancestral relationships among the species or the populations of the considered individuals. In particular, we show that when gene trees and species trees have a particular caterpillarshaped matching topology t of size n, the approximations of the value of \(i^*\) determined in Sect. 3 for the paths in \({\mathcal {D}}_n,{\mathcal {D}}_n^{\prime },\) and \({\mathcal {D}}_n^{\prime \prime }\) can assist in locating the position of the speciation event \(\sigma\) over t—the splitting of a leaf node of t into two children nodes—for which the resulting tree \(t_{\sigma }\) has the largest increase in the number of coalescent histories with respect to t.
2 Standard and weighted Dyck paths
3 Heights in \({\mathcal {D}}_n, {\mathcal {D}}_n^{\prime },\) and \({\mathcal {D}}_n^{\prime \prime }\)
In the following, we write \(f(n) \approx {\tilde{f}}(n)\) when theoretical arguments coupled with numerical calculations indicate that, for n sufficiently large, the ratio \(f(n)/{\tilde{f}}(n)\) is “close” to 1. When \(f(n) / {\tilde{f}}(n)\) converges to 1, we write \(f(n) \sim {\tilde{f}}(n)\).
3.1 Heights in \({\mathcal {D}}_n\)
Although Dyck paths have been studied by several authors, up to our knowledge results for the mean of the height \(y_i\) do not explicitly appear in the literature for arbitrary values of \(i \in [1,n]\). In this section, we fill this gap by determining closed formulas for the expectation \({\mathrm{E}}_n(y_i)\) and for the second moment \({\mathrm{E}}_n(y_i^2)\) of the random variable \(y_i\), when paths of \({\mathcal {D}}_n\) are selected uniformly at random. We find that \({\mathrm{m}}(n)\approx 2\sqrt{n/\pi }\) and \(i^*(n) \approx n/2\). In Sect. 3.1.1, we provide a formula for the joint probability \({\mathrm{Prob}}( y_i=q \, \& \, y_j=p )\), which enables the computation of the expectation \({\mathrm{E}}_n(y_jy_i=q)\) of the height \(y_j\) in a random path of size n conditioning on a given a value q for the height \(y_i\). Our calculations also relate to the study of the Brownian excursion process \(\{ e(t) : t \in [0,1] \}\), which is Brownian motion conditioned to be 0 at \(t=0\) and \(t=1\) and positive in the interior. Indeed, if y(i) denotes the height at position \(1 \le i \le 2n\) of a random Dyck path of semilength n, then \(y_i = y(2i  y_i)\), and it is a known fact [18] that for a fixed \(t \in [0,1]\) the sequence of variables \(y(2 n t)/\sqrt{2n}\) converges for \(n \rightarrow \infty\) to the random variable e(t).
Remark
Note that, as reported e.g. in [13] (Proposition V.4), the expected height of a random Dyck path of size n measured at its highest point is asymptotically \(\sqrt{\pi n}\). Up to a constant factor \(\sqrt{\pi }/(2\sqrt{1/\pi })= \pi /2\), this expectation corresponds to the estimate given in (10) for expected height of a random Dyck path of size n measured at position \(i^*(n)\).
3.1.1 Joint and conditional probability of \(y_i\) and \(y_j\)
3.2 Heights in \({\mathcal {D}}_n^{\prime }\)
In this section, we study the height variables \(y_i\) for paths of \({\mathcal {D}}_n^{\prime }\). Note that, as described in Fig. 2, paths of \({\mathcal {D}}_n^{\prime }\) are in bijective correspondence with interconnection networks over 2n points. In particular, for a given path of \({\mathcal {D}}_n^{\prime }\) the value of \(y_i\) determines the number of arcs that are active (open) in the associated interconnection network at the position where the ith arc opens. Here, we determine exact formulas for the expectation \({\mathrm{E}}_n(y_i)\) and for the second moment \({\mathrm{E}}_n(y_i^2)\) of the random variable \(y_i\). In particular, we find that \({\mathrm{m}}(n) \approx n/2\) and \(i^*(n) \approx 3n/4\), while the expected length of the last descent is \({\mathrm{E}}_n(y_n) = 1 + 4^n/{{2n}\atopwithdelims (){n}} \sim \sqrt{\pi \, n}\).
Remark
Equation (27) indicates that asymptotically the ratio \({\mathrm{m}}(n)/n\) converges to a value close—if not equal—to 1/2. Interestingly, the expected height—the ordinate of the highest point—of a random uniform path in \({\mathcal {D}}_n^{\prime }\) grows like n/2. Indeed, the height of a path in \({\mathcal {D}}_n^{\prime }\) corresponds to the width of the interconnection network over 2n points associated with the path (Fig. 2), that is, to the maximum number of arcs met by a vertical line intersecting the diagram of the network. As reported in [13] (see also references therein), the expected width of a random interconnection network over 2n points grows asymptotically like n/2.
3.3 Heights in \({\mathcal {D}}_n^{\prime \prime }\)
The product \(d_{n,i,q}^{\prime \prime }=d_{i,q}^{\prime \prime } \cdot s_{ni,q}^{\prime \prime }\) gives the number of paths in \({\mathcal {D}}_n^{\prime \prime }\) such that \(y_i=q\), and the related probability can be computed as \({\mathrm{Prob}}(y_i=q) = d_{n,i,q}^{\prime \prime }/{\mathcal {D}}_n^{\prime \prime }\). As depicted in Fig. 8 (right), numerical calculations indicate that for a fixed i the distribution \({\mathrm{Prob}}(y_i=q)\) is unimodal and such that values of q of nonnegligible probability are concentrated around the mode \(\mu _n(y_i)\), the latter being thus close to the mean, \(\mu _n(y_i) \approx {\mathrm{E}}_n(y_i)\).
Remark
3.3.1 Last descent \(y_n\)
3.4 Estimates found for the value of \({\mathrm{E}}_n(y_n)\), \({\mathrm{m}}(n)\) and \(i^*(n)\)
Asymptotic values of \({\text{E}}_n(y_n), {\text{m}}(n)\) and \(i^*(n)\) found for paths in \({\mathcal {D}}_n\) (first row), \({\mathcal {D}}_n^{\prime }\) (second row), and \({\mathcal {D}}_n^{\prime \prime }\) (third row)
\({\text{E}}_n(y_n)\)  \({\text{m}}(n)\)  \(i^*(n)\)  

\({\mathcal {D}}_n\)  \(\sim 3\)  \(\approx 2\sqrt{\frac{n}{\pi }}\)  \(\approx \frac{n}{2}\) 
\({\mathcal {D}}_n^{\prime }\)  \(\sim \sqrt{\pi n}\)  \(\approx \frac{n}{2}\)  \(\approx \frac{3\,n}{4}\) 
\({\mathcal {D}}_n^{\prime \prime }\)  \(\approx \frac{3 \, n^{2/3}}{2}\)  \(\approx \frac{(1+\sqrt{17})\,n}{8}\)  \(\approx \frac{(9+\sqrt{17})\,n}{16}\) 
4 Relationships with the variability of the number of coalescent histories
Some of the results presented in the previous sections relate to the study of the variability of the number of coalescent histories [7, 10, 11, 23, 24], structures used in phylogenetic models of evolution for representing the combinatorially different configurations that a gene tree topology—representing how genes sampled from individuals have evolved from a common ancestor—can assume along the branches of a species tree—which describes the evolutionary relationships among the species or the populations of the considered individuals (Fig. 10). These structures are used in fundamental calculations of gene tree probabilities (see e.g. [6, 7]) for the inference of species trees based on collections of gene trees derived from genetic data. As the cost of these calculations is strongly affected by the number of coalescent histories possible for a gene tree G and a species tree S, it is important to characterize those pairs (G, S) for which the number of coalescent histories is larger or smaller.
4.1 Coalescent histories for three families of caterpillarshaped trees
Under the matching assumption \(G=S=t\), coalescent histories of a caterpillarshaped tree t (Fig. 11a) can be interpreted as Dyck paths with colored up steps. In particular, here we focus on coalescent histories for three families of caterpillarshaped trees that have been already considered in the literature [7, 10], namely the caterpillar, the lodgepole, and the pitchfork family (Fig. 11: b, c, and d respectively).
If t is a caterpillar tree of size n (Fig. 11b), then there exists a bijective correspondence between the coalescent histories of t and the Dyck paths of size n: each colaescent history h of t is encoded by the Dyck path \(\gamma _h \in {\mathcal {D}}_n\) such that the value of \(y_i\)—the height of the ith up step \(U_i\)—in \(\gamma _h\) equals the depth of the coalescent event \(k_i\) in h, in symbols \(y_i(\gamma _h) = {\text {depth}}_{k_i}(h)\). For instance, the Dyck path \(\gamma _h \in {\mathcal {D}}_4\) associated with the coalescent history depicted in Fig. 11b is \(\gamma _h=UDUUDUDD\), where \((y_1,y_2,y_3,y_4)=(1,1,2,2)\) as determined by the depth of the coalescent events \(k_1,k_2,k_3,\) and \(k_4\) under the considered coalescent history.
When t is a lodgepole tree of size n (Fig. 11c), the set of coalescent histories of t is in bijection with the set \({\mathcal {D}}_n^{\ell }\) of labeled Dyck paths of size n such that each up step \(U_i\) of a path \(\gamma \in {\mathcal {D}}_n^{\ell }\) is colored by an integer label in the range \([1,y_i+1]\), where \(y_i\) is the height of \(U_i\) in \(\gamma\). More precisely, a coalescent history h of t (Fig. 11c) is encoded by the path \(\gamma _h \in {\mathcal {D}}_n^{\ell }\) where (1) as in the caterpillar case \(y_i(\gamma _h) = {\text {depth}}_{k_i}(h)\), and (2) the up step \(U_i\) in \(\gamma _h\) is colored with an integer label whose value corresponds to the depth under h of the cherry node descending from node \(k_i\) in t. For example, the path \(\gamma _h \in {\mathcal {D}}_4^{\ell }\) associated with the history h depicted in Fig. 11c is \(\gamma _h=UDUUDUDD\) with labels for its up steps \((U_1,U_2,U_3,U_4)\) given by (2, 1, 2, 3). Note that for every history h of t the depth of the cherry node descending from \(k_i\) exceeds at most by one the depth of node \(k_i\). Therefore, for every h the label for the up step \(U_i\) of \(\gamma _h\) is in the range \([1,y_i+1]\) as required for paths in \({\mathcal {D}}_n^{\ell }\). Furthermore, observe that \({\mathcal {D}}_n^{\ell }\) resembles \({\mathcal {D}}_n^{\prime }\) in the sense that both sets are derived from Dyck paths by assigning to up steps U a weight, \(y(U)+1\) and y(U), that is linear in the height y(U). In particular, paths in \({\mathcal {D}}_n^{\ell }\) correspond bijectively to indecomposable paths of \({\mathcal {D}}_{n+1}^{\prime }\) [10], and the cardinalities \({\mathcal {D}}_n^{\ell },{\mathcal {D}}_n^{\prime }\) are asymptotically equivalent, \({\mathcal {D}}_n^{\ell } \sim {\mathcal {D}}_n^{\prime }\).
Finally, if t is a pitchfork tree of size n (Fig. 11d), then the set of coalescent histories of t corresponds bijectively to the set \({\mathcal {D}}_n^{p}\) of labeled Dyck paths of size n in which each up step \(U_i\) of \(\gamma \in {\mathcal {D}}_n^{p}\) is colored by an integer label in the range \([1,2+5y_i/2+y_i^2/2\)], where \(y_i\) is measured in \(\gamma\). Similarly to the lodgepole case, we associate a history h of t (Fig. 11d) with the path \(\gamma _h \in {\mathcal {D}}_n^{p}\) such that (1) \(y_i(\gamma _h) = {\text {depth}}_{k_i}(h)\), and (2) the up step \(U_i\) in \(\gamma _h\) is colored with an integer label whose value determines (as detailed below) the depth under h of the two internal nodes \(k_{i,1}, k_{i,2}\) of the subtree s appended to \(k_i\) in t. More in details, assuming without loss of generality \(k_{i,2}\) below \(k_{i,1}\) in t, in (2) we define the label for the ith up step \(U_i\) of \(\gamma _h\) as the position (first, second, and so on) under the lexicographic order of the pair of integers \(\big ({\text {depth}}_{k_{i,1}}(h),{\text {depth}}_{k_{i,2}}(h)\big )\) in the set \(A(y_i)=\{(a_{i,1},a_{i,2}): 1\le a_{i,1} \le y_i+1 \text { and } 1\le a_{i,2} \le a_{i,1}+1 \}\) of cardinality \(A(y_i)=2+5y_i/2+y_i^2/2\). For instance, the path \(\gamma _h \in {\mathcal {D}}_4^{p}\) that corresponds to the history h depicted in Fig. 11D is \(\gamma _h=UDUUDUDD\) with labels for its up steps \((U_1,U_2,U_3,U_4)\) given by (5, 1, 3, 8), where e.g. label 3 for step \(U_3\) comes from the fact that the pair \(\big ({\text {depth}}_{k_{3,1}}(h),{\text {depth}}_{k_{3,2}}(h)\big )=(2,1)\) is in position 3 in the set \(A(y_3)=A(2)=\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(3,4)\}\). Note that both \({\mathcal {D}}_n^{p}\) and \({\mathcal {D}}_n^{\prime \prime }\) derive from Dyck paths by assigning to up steps U a weight that is quadratic in the height y(U).
4.1.1 Variability of the number of coalescent histories and values of \(i^*\)
These empirical results can be explained by taking a closer look at the branching structure of the trees under consideration. The idea is that, extending a branch at position i of a caterpillar, lodgepole or pitchfork tree t, the magnitude of the increase in the number of coalescent histories depends on the depth of the coalescent event \(k_i\) for the coalescent histories of t: a deeper coalescent event \(k_i\) creates more possibilities for the mapping of the nodes of the left subtree of \(k_i\) (Fig. 11). This observation, coupled with (47), gives the intuition for the larger increase when the extended branch is taken at positions i close to \(i^*\).
5 Conclusions

Refine the empirical estimates reported in Sect. 3.4. Note that the study of the height variables \(y_i\) in random paths of \({\mathcal {D}}_n^{\prime \prime }\) is a problem for which it seems quite hard to obtain exact solutions. Indeed, even the number of paths \({\mathcal {D}}_n^{\prime \prime }\) is a quantity accessible only by asymptotic approximations.

Extend our calculations to consider models of Dyck paths in which the ith up step is assigned a weight \(y_i^w\), for a given positive integer w. When the size n is sufficiently large, preliminary results indicate that the mean length of the last descent is of order \(n^{w/(w+1)}\).

Investigate in more detail the relationships between the statistic \(y_n\)—the length of the last descent in weighted Dyck paths—and families of polynomials. Here, we have shown that the number of paths in \({\mathcal {D}}_n^{\prime \prime }\) with a fixed value of \(y_n\) can be expressed by combining Euler numbers and Meixner polynomials.
Notes
Acknowledgements
Support was provided by a LeviMontalcini grant to FD from the Ministero dell’Istruzione, dell’Università e della Ricerca.
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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