We give the generating function of split $(n+t)$-colour partitions and obtain an analogue of Euler’s identity for split $n$-colour partitions. We derive a combinatorial relation between the number of restricted split $n$-colour partitions and the function $\unicode[STIX]{x1D70E}_{k}(\unicode[STIX]{x1D707})=\sum _{d|\unicode[STIX]{x1D707}}d^{k}$. We introduce a new class of split perfect partitions with $d(a)$ copies of each part $a$ and extend the work of Agarwal and Subbarao [‘Some properties of perfect partitions’, Indian J. Pure Appl. Math 22(9) (1991), 737–743].

In this paper a determinant identity is established, from which a simple proof of the multivariate Lagrange–Good inversion formula follows directly. Further discussion on a discrete analogue of the Lagrange–Good inversion formula is also presented.

In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.

We study the coefficients of algebraic functions ∑n≥0fnzn. First, we recall the too-little-known fact that these coefficients fn always admit a closed form. Then we study their asymptotics, known to be of the type fn ~ CAnnα. When the function is a power series associated to a context-free grammar, we solve a folklore conjecture: the critical exponents α cannot be 1/3 or −5/2; they in fact belong to a proper subset of the dyadic numbers. We initiate the study of the set of possible values for A. We extend what Philippe Flajolet called the Drmota–Lalley–Woods theorem (which states that α=−3/2 when the dependency graph associated to the algebraic system defining the function is strongly connected). We fully characterize the possible singular behaviours in the non-strongly connected case. As a corollary, the generating functions of certain lattice paths and planar maps are not determined by a context-free grammar (i.e., their generating functions are not ℕ-algebraic). We give examples of Gaussian limit laws (beyond the case of the Drmota–Lalley–Woods theorem), and examples of non-Gaussian limit laws. We then extend our work to systems involving non-polynomial entire functions (non-strongly connected systems, fixed points of entire functions with positive coefficients). We give several closure properties for ℕ-algebraic functions. We end by discussing a few extensions of our results (infinite systems of equations, algorithmic aspects).

We generalize and improve recent results by Bóna and Knopfmacher and by Banderier and Hitcz-enko concerning the joint distribution of the sum and number of parts in tuples of restricted compositions. Specifically, we generalize the problem to general combinatorial classes and relax the requirement that the sizes of the compositions be equal. We extend the main explicit results to enumeration problems whose counting sequences are Riordan arrays. In this framework, we give an alternative method for computing asymptotics in the supercritical case, which avoids explicit diagonal extraction.

Let ${\cal A}$ be a minor-closed class of labelled graphs, and let ${\cal G}_{n}$ be a random graph sampled uniformly from the set of n-vertex graphs of ${\cal A}$. When n is large, what is the probability that ${\cal G}_{n}$ is connected? How many components does it have? How large is its biggest component? Thanks to the work of McDiarmid and his collaborators, these questions are now solved when all excluded minors are 2-connected.

Using exact enumeration, we study a collection of classes ${\cal A}$ excluding non-2-connected minors, and show that their asymptotic behaviour may be rather different from the 2-connected case. This behaviour largely depends on the nature of the dominant singularity of the generating function C(z) that counts connected graphs of ${\cal A}$. We classify our examples accordingly, thus taking a first step towards a classification of minor-closed classes of graphs. Furthermore, we investigate a parameter that has not received any attention in this context yet: the size of the root component. It follows non-Gaussian limit laws (Beta and Gamma), and clearly merits a systematic investigation.

We prove that the number of 1324-avoiding permutations of length n is less than $(7+4\sqrt{3})^n$. The novelty of our method is that we injectively encode such permutations by a pair of words of length n over a finite alphabet that avoid a given factor.

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.

In a 1976 paper published in Science, Knuth presented an algorithm to sample (non-uniform) self-avoiding walks crossing a square of side k. From this sample, he constructed an estimator for the number of such walks. The quality of this estimator is directly related to the (relative) variance of a certain random variable Xk. From his experiments, Knuth suspected that this variance was extremely large (so that the estimator would not be very efficient). But how large? For the analogous Rosenbluth algorithm, which samples unconfined self-avoiding walks of length n, the variance of the corresponding estimator is believed to be exponential in n.

A few years ago, Bassetti and Diaconis showed that, for a sampler à la Knuth that generates walks crossing a k × k square and consisting of North and East steps, the relative variance is only $O(\sqrt k)$. In this note we take one step further and show that, for walks consisting of North, South and East steps, the relative variance jumps to $2^{k(k+1)}/(k+1)^{2k}$. This is exponential in the average length of the walks, which is of order k2. We also obtain partial results for general self-avoiding walks crossing a square, suggesting that the relative variance could be exponential in k2 (which is again the average length of these walks).

Knuth's algorithm is a basic example of a widely used technique called sequential importance sampling. The present paper, following the paper by Bassetti and Diaconis, is one of very few examples where the variance of the estimator can be found.

A colouring of the vertices of a regular polygon is symmetric if it is invariant under some reflection of the polygon. We count the number of symmetric $r$-colourings of the vertices of a regular $n$-gon.

An $r$-ary necklace (bracelet) of length $n$ is an equivalence class of $r$-colourings of vertices of a regular $n$-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric $r$-ary necklaces (bracelets) of length $n$ is $\frac{1}{2} (r+ 1){r}^{n/ 2} $ if $n$ is even, and ${r}^{(n+ 1)/ 2} $ if $n$ is odd.

This paper is the first application of the compensation approach (a well-established theory in probability theory) to counting problems. We discuss how this method can be applied to a general class of walks in the quarter plane +2 with a step set that is a subset of

\[ \{(-1,1),(-1,0),(-1,-1),(0,-1),(1,-1)\}\]

Hu et al. [“A boundary problem for group testing”, SIAM J. Algebraic Discrete Meth.2 (1981), 81–87] conjectured that the minimax test number to find d defectives in 3d items is 3d−1, a surprisingly difficult combinatorial problem about which very little is known. In this article we state three more conjectures and prove that they are all equivalent to the conjecture of Hu et al. Notably, as a byproduct, we also obtain an interesting upper bound for M(d,n).