Slit-slide-sew bijections for constellations and quasiconstellations

TL;DR

This work extends slit-slide-sew bijections to the realm of constellations and quasiconstellations by introducing an orientation-aware involution on hypermaps with distinguished elements. It develops a canonical slitting path and sewing procedure that reorients the map and transfers degree between faces, yielding exact counting identities for face-degree distributions and a uniform sampling algorithm for prescribed types. The methodology hinges on distinguishing elements, geodesic-driven slits, and a growth framework that builds complex constellations from a base p-ary-tree structure, connecting to classical enumeration results and Hurwitz-type considerations. The results provide both theoretical bijections and a practical tool for generating random constellations/quasiconstellations with specified face degrees, with implications for permutation factorizations and related enumerative questions.

Abstract

We extend so-called slit-slide-sew bijections to constellations and quasiconstellations. We present an involution on the set of hypermaps given with an orientation, one distinguished corner, and one distinguished edge leading away from the corner while oriented in the given orientation. This involution reverts the orientation, exchanges the distinguished corner with the distinguished edge in some sense, slightly modifying the degrees of the incident faces in passing, while keeping all the other faces intact. The construction consists in building a canonical path from the distinguished elements, slitting the map along it, and sewing back after sliding by one unit along the path. The involution specializes into a bijection interpreting combinatorial identities linking the numbers of constellations or quasiconstellations with a given face degree distribution, where the degree distributions differ by one $+1$ and one $-1$. In particular, this allows to recover the counting formula for constellations or quasiconstellations with a given face degree distribution. Our bijections furthermore provide an algorithm for sampling a hypermap uniformly distributed among constellations or quasiconstellations with prescribed face degrees.

Figures (18)

• Figure 1: A quasi-$3$-constellation of type $(9,6,2,6,3,3,4,3,3)$. The two flawed faces are $f_3$ and $f_7$; throughout the paper, we highlight flawed faces in orange. Every light face has a marked corner, always represented by a red arrowhead.
• Figure 2: A quasi-$3$-constellation of type $(9,6,2,6,3,3,4,3,3)$. The two flawed faces are $f_3$ and $f_7$; throughout the paper, we highlight flawed faces in orange. Every light face has a marked corner, always represented by a red arrowhead.
• Figure 3: A quasi-$2$-constellation and the corresponding quasibipartite map, obtained by squeezing all the dark faces.
• Figure 4: Bijection between $p$-constellations of type $(pn)$ and $p$-ary trees with $n$ nodes. Each dark face corresponds to a $p$-ary node and each vertex to a leaf. We follow the contour of the light face, starting from its marked corner, in a given orientation, say with the light face on the left. Each time we encounter an unvisited dark face, say $D$, we consider the $3$ incident vertices, say $v_1$, $v_2$, $v_3$. For $1\le i \le 3$, we link the node corresponding to $D$ to the node corresponding to the left-most unvisited dark face incident to $v_i$ or, by default, to the leaf corresponding to $v_i$. We root the tree at the corner following the marked corner of $f_1$.
• Figure 5: Growing a $3$-constellation of type $(6,3,1,11)$. Step 0. We start with a random $3$-constellation of type $(21)$, thus having a unique (purple) light face, which we think of an external face. Step 1. Randomly selecting a (green) vertex and a (blue) edge, our bijections allow to add a new degree-$1$ (yellow) light face at the cost of removing one degree from the external face. The resulting quasi-$3$-constellation comes with a distinguished (blue) corner in the external face, which we forget for the next step. Step 2--6. We repeat the process of randomly transferring degrees from the external light face to the yellow face, thus obtaining a random $3$-constellation of type $(6,15)$. Step 7--9. We add a new (green) degree-$1$ light face then "inflate" it to obtain a random $3$-constellation of type $(6,3,12)$. Step 10. Finally, we add a new (blue) degree-$1$ light face to obtain a random quasi-$3$-constellation of type $(6,3,1,11)$. In this final step, the selected (green) vertex and (blue) edge used for applying the bijection are also represented.

Theorems & Definitions (14)

• Proposition 1: Transferring one degree from $f_1$ to $f_2$
• Remark 1
• Proposition 2: Transferring the degree of a degree $1$-face $f_1$ to $f_2$
• Proposition 3
• Corollary 4
• proof
• proof : Proof of Proposition \ref{['propirreg']}
• Theorem 5
• proof
• Claim