Conventions fixed on this page

The page uses two step pictures and treats them as one subject only through their proved dictionary. Lattice paths, step sets and step words is the ambient definition, Dyck paths of semilength $n$ fixes the diagonal picture, and The two step sets describe the same objects: $U\mapsto N$, $D\mapsto E$ is a bijection matching the diagonal $y=x$ with the level $0$ is the only place where the monotone picture is identified with it.

The indexing starts at $0$. Dyck paths have semilength $n$, the Catalan number $C_n$ counts semilength-$n$ paths by The Catalan number $C_n:=\mid {\mathcal{D}}_n\mid$, and the base case is $C_0=1$. Every count on the page is written with that convention visible rather than hidden inside a later formula.

"Strictly above" and "weakly above" are different conditions and are never merged. The reflection principle counts paths staying strictly above a level; the Dyck-path count uses weakly above because the path may touch height $0$; and The cycle lemma (Dvoretzky–Motzkin): if every $a_i\le 1$ and $\parallel a\parallel =k\ge 1$, then exactly $k$ of the $m$ cyclic shifts of $a$ have all partial sums positive fixes the orientation that a good shift is one whose partial sums are all strictly positive, with shifts indexed by their starting position.

A bijection on this page is always given with a two-sided inverse. That convention is what blocks the companion page's false bijection, and it is why the tail-swap of Tail-swapping is a sign-reversing involution on the intersecting systems is stated as an involution rather than as a cancellation slogan.

Two familiar refinements are left out because this page does not build the extra machinery they need. The Hankel determinant identity $\det (C_{i+j}{)}_{0\le i,j\le n-1}=1$ would need $\det (\mid \mathcal{M}(A_i,E_j)\mid {)}_{i,j}={\sum }_{\pi \in S_r}\mathrm{sgn}(\pi )\cdot \mathrm{\#}\{\text{non-intersecting }\pi \text{-systems}\}$ applied to a path family closed under the same tail-swap, together with a theorem counting the monotone paths that stay weakly below a fixed diagonal. The Narayana refinement counts Dyck paths by their number of peaks, and none of the routes built here tracks that statistic. Both are therefore recorded as not built here, and Huq §2.5 is the source in hand for the second.