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We call $i$ a fixed point of a given sequence if the value of that sequence at the $i$-th position coincides with $i$. Here, we enumerate fixed points in the class of restricted growth sequences. The counting process is conducted by calculation of generating functions and leveraging a probabilistic sampling method.
We study the distributional properties of horizontal visibility graphs associated with random restrictive growth sequences and random set partitions of size $n.$ Our main results are formulas expressing the expected degree of graph nodes in terms of
In this paper we study a subfamily of a classic lattice path, the emph{Dyck paths}, called emph{restricted $d$-Dyck} paths, in short $d$-Dyck. A valley of a Dyck path $P$ is a local minimum of $P$; if the difference between the heights of two consecu
We count orientations of $G(n,p)$ avoiding certain classes of oriented graphs. In particular, we study $T_r(n,p)$, the number of orientations of the binomial random graph $G(n,p)$ in which every copy of $K_r$ is transitive, and $S_r(n,p)$, the number
A restricted growth function (RGF) of length n is a sequence w = w_1 w_2 ... w_n of positive integers such that w_1 = 1 and w_i is at most 1 + max{w_1,..., w_{i-1}} for i at least 2. RGFs are of interest because they are in natural bijection with set
The greedy tree $mathcal{G}(D)$ and the $mathcal{M}$-tree $mathcal{M}(D)$ are known to be extremal among trees with degree sequence $D$ with respect to various graph invariants. This paper provides a general theorem that covers a large family of inva