Abstract:
Consider a rooted binary tree with n nodes. Assign with the root the abscissa 0, and with the left (resp. right) child of a node of abscissa i the abscissa i-1 (resp. i+1). We prove that the number of binary trees of size n having exactly n_i nodes at abscissa i, for l =< i =< r (with n = sum_i n_i), is $$ \frac{n_0}{n_l n_r} {{n_{-1}+n_1} \choose {n_0-1}} \prod_{l\le i\le r \atop i\not = 0}{{n_{i-1}+n_{i+1}-1} \choose {n_i-1}}, $$ with n_{l-1}=n_{r+1}=0. The sequence (n_l, ..., n_{-1};n_0, ..., n_r) is called the vertical profile of the tree. The vertical profile of a uniform random tree of size n is known to converge, in a certain sense and after normalization, to a random mesure called the integrated superbrownian excursion, which motivates our interest in the profile. We prove similar looking formulas for other families of trees whose nodes are embedded in Z. We also refine these formulas by taking into account the number of nodes at abscissa j whose parent lies at abscissa i, and/or the number of vertices at abscissa i having a prescribed number of children at abscissa j, for all i and j. Our proofs are bijective.

Abstract:
High-latitude coral reefs may be a refuge and area of reef expansion under climate change. As these locations are expected to become dryer and as livestock and agricultural yields decline, coastal populations may become increasingly dependent on marine resources. To evaluate this social–ecological conundrum, we examined the Grand Récif of Toliara (GRT), southwest Madagascar, which was intensively studied in the 1960s and has been highly degraded since the 1980s. We analyzed the social and ecological published and unpublished literature on this region and provide new data to assess the magnitude of the changes and evaluate the causes of reef degradation. Top-down controls were identified as the major drivers: human population growth and migrations, overfishing, and climate change, specifically decreased rainfall and rising temperature. Water quality has not changed since originally studied, and bottom-up control was ruled out. The identified network of social–ecological processes acting at different scales implies that decision makers will face complex problems that are linked to broader social, economic, and policy issues. This characterizes wicked problems, which are often dealt with by partial solutions that are exploratory and include inputs from various stakeholders along with information sharing, knowledge synthesis, and trust building. A hybrid approach based on classical fishery management options and preferences, along with monitoring, feedback and forums for searching solutions, could move the process of adaptation forward once an adaptive and appropriately scaled governance system is functioning. This approach has broad implications for resources management given the emerging climate change and multiple social and environmental stresses.

Abstract:
An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^{m}, which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S_n on these spaces is conjectured to be closely related to the natural representation of S_n on (labelled) intervals of the m-Tamari lattice, which we study in this paper. An interval [P,Q] of T_n^{m} is labelled if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S_n acts on labelled intervals of T_n^{m} by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S_n. In particular, the dimension of the representation, that is, the number of labelled m-Tamari intervals of size n, is found to be (m+1)^n(mn+1)^{n-2}. These results are new, even when m=1. The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.

Abstract:
An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0,0), ending at (mn,n), and never going below the line {x=my. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T_n^(m), which generalizes the usual Tamari lattice T_n obtained when m=1. This lattice was introduced by F. Bergeron in connection with the study of coinvariant spaces. He conjectured several intriguing formulas dealing with the enumeration of intervals in this lattice. One of them states that the number of intervals in T_n^(m) is $$ \frac {m+1}{n(mn+1)} {(m+1)^2 n+m\choose n-1}. $$ This conjecture was proved recently, but in a non-bijective way, while its form strongly suggests a connection with plane trees. Here, we prove another conjecture of Bergeron, which deals with the number of labelled, intervals. An interval [P,Q] of T_n^(m) is labelled, if the north steps of Q are labelled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. We prove that the number of labelled intervals in T_n^(m) is $$ {(m+1)^n(mn+1)^{n-2}}. $$ The form of these numbers suggests a connection with parking functions, but our proof is non-bijective. It is based on a recursive description of intervals, which translates into a functional equation satisfied by the associated generating function. This equation involves a derivative and a divided difference, taken with respect to two additional variables. Solving this equation is the hardest part of the paper. Finding a bijective proof remains an open problem.

Abstract:
Regeneration of artificially induced lesions was monitored in nubbins of the branching coral Acropora muricata at two reef-flat sites representing contrasting environments at Réunion Island (21°07′S, 55°32′E). Growth of these injured nubbins was examined in parallel, and compared to controls. Biochemical compositions of the holobiont and the zooxanthellae density were determined at the onset of the experiment, and the photosynthetic efficiency (Fv/Fm) of zooxanthellae was monitored during the experiment. Acropora muricata rapidly regenerated small lesions, but regeneration rates significantly differed between sites. At the sheltered site characterized by high temperatures, temperature variations, and irradiance levels, regeneration took 192 days on average. At the exposed site, characterized by steadier temperatures and lower irradiation, nubbins demonstrated fast lesion repair (81 days), slower growth, lower zooxanthellae density, chlorophyll a concentration and lipid content than at the former site. A trade-off between growth and regeneration rates was evident here. High growth rates seem to impair regeneration capacity. We show that environmental conditions conducive to high zooxanthellae densities in corals are related to fast skeletal growth but also to reduced lesion regeneration rates. We hypothesize that a lowered regenerative capacity may be related to limited availability of energetic and cellular resources, consequences of coral holobionts operating at high levels of photosynthesis and associated growth.

Abstract:
We consider planar lattice walks that start from (0,0), remain inthe first quadrant i, j >= 0, and are made of three types of steps: North-East, West and South. These walks are known to have remarkable enumerative and probabilistic properties: -- they are counted by nice numbers (Kreweras 1965), -- the generating function of these numbers is algebraic (Gessel 1986), -- the stationary distribution of the corresponding Markov chain in the quadrant has an algebraic probability generating function (Flatto and Hahn 1984). These results are not well understood, and have been established via complicated proofs. Here we give a uniform derivation of all of them, whichis more elementary that those previously published.We then go further by computing the full law of the Markov chain. This helps to delimit the border of algebraicity: the associated probability generating function is no longer algebraic, unless a diagonal symmetry holds. Our proofs are based on the solution of certain functional equations,which are very simple to establish. Finding purely combinatorial proofs remains an open problem.

Abstract:
We study three families of labelled plane trees. In all these trees, the root is labelled 0, and the labels of two adjacent nodes differ by $0, 1$ or -1. One part of the paper is devoted to enumerative results. For each family, and for all $j\in \ns$, we obtain closed form expressions for the following three generating functions: the generating function of trees having no label larger than $j$; the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labelled $j$; and finally the (bivariate) generating function of trees, counted by the number of edges and the number of nodes labelled at least $j$. Strangely enough, all these series turn out to be algebraic, but we have no combinatorial intuition for this algebraicity. The other part of the paper is devoted to deriving limit laws from these enumerative results. In each of our families of trees, we endow the trees of size $n$ with the uniform distribution, and study the following random variables: $M\_n$, the largest label occurring in a (random) tree; $X\_n(j)$, the number of nodes labelled $j$; and $X\_n^+(j)$, the number of nodes labelled $j$ or more. We obtain limit laws for scaled versions of these random variables. Finally, we translate the above limit results into statements dealing with the integrated superBrownian excursion (ISE). In particular, we describe the law of the supremum of its support (thus recovering some earlier results obtained by Delmas), and the law of its distribution function at a given point. We also conjecture the law of its density (at a given point).

Abstract:
It is well-known that the length generating function E(t) of Dyck paths (excursions with steps +1 and -1) satisfies 1-E+t^2E^2=0. The generating function E^(k)(t) of Dyck paths of height at most k is E^(k)=F_k/F_{k+1}, where the F_k are polynomials in t given by F_0=F_1=1 and F_{k+1}= F_k-t^2F_{k-1}. This means that the generating function of these polynomials is \sum_{k\ge 0} F_k z^k= 1/(1-z+t^2z^2). We note that the denominator of this fraction is the minimal polynomial of the algebraic series E(t). This pattern persists for walks with more general steps. For any finite set of steps S, the generating function E^(k)(t) of excursions (generalized Dyck paths) taking their steps in S and of height at most k is the ratio F_k/F_{k+1} of two polynomials. These polynomials satisfy a linear recurrence relation with coefficients in Q[t]. Their (rational) generating function can be written \sum_{k\ge 0} F_k z^k= N(t,z)/D(t,z). The excursion generating function E(t) is algebraic and satisfies D(t,E(t))=0 (while N(t,E(t))\not = 0). If max S=a and min S=b, the polynomials D(t,z) and N(t,z) can be taken to be respectively of degree d_{a,b}=binomial(a+b,a) and d_{a,b}-a-b in z. These degrees are in general optimal: for instance, when S={a,-b} with a and b coprime, D(t,z) is irreducible, and is thus the minimal polynomial of the excursion generating function E(t). The proofs of these results involve a slightly unusual mixture of combinatorial and algebraic tools, among which the kernel method (which solves certain functional equations), symmetric functions, and a pinch of Galois theory.

Abstract:
A self-avoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Pr\'ea in 1997. He defined 4 classes of prudent walks, of increasing generality, and wrote a system of recurrence relations for each of them . However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is well-known to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even D-finite. The fourth class -- general prudent walks -- is the only isotropic one, and still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be non-D-finite. We also study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed.

Abstract:
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers a(n), but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions.