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Enumeration of planar bipartite tight irreducible maps

Jérémie Bouttier, Emmanuel Guitter, Hugo Manet

Abstract

We consider planar bipartite maps which are both tight, i.e. without vertices of degree $1$, and $2b$-irreducible, i.e. such that each cycle has length at least $2b$ and such that any cycle of length exactly $2b$ is the contour of a face. It was shown by Budd that the number $\mathcal N_n^{(b)}$ of such maps made out of a fixed set of $n$ faces with prescribed even degrees is a polynomial in both $b$ and the face degrees. In this paper, we give an explicit expression for $\mathcal N_n^{(b)}$ by a direct bijective approach based on the so-called slice decomposition. More precisely, we decompose any of the maps at hand into a collection of $2b$-irreducible tight slices and a suitable two-face map. We show how to bijectively encode each $2b$-irreducible slice via a $b$-decorated tree drawn on its derived map, and how to enumerate collections thereof. We then discuss the polynomial counting of two-face maps, and show how to combine it with the former enumeration to obtain $\mathcal N_n^{(b)}$.

Enumeration of planar bipartite tight irreducible maps

Abstract

We consider planar bipartite maps which are both tight, i.e. without vertices of degree , and -irreducible, i.e. such that each cycle has length at least and such that any cycle of length exactly is the contour of a face. It was shown by Budd that the number of such maps made out of a fixed set of faces with prescribed even degrees is a polynomial in both and the face degrees. In this paper, we give an explicit expression for by a direct bijective approach based on the so-called slice decomposition. More precisely, we decompose any of the maps at hand into a collection of -irreducible tight slices and a suitable two-face map. We show how to bijectively encode each -irreducible slice via a -decorated tree drawn on its derived map, and how to enumerate collections thereof. We then discuss the polynomial counting of two-face maps, and show how to combine it with the former enumeration to obtain .

Paper Structure

This paper contains 34 sections, 27 theorems, 68 equations, 21 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Basic definitions.
  4. Acknowledgements.
  5. Slice decomposition of (tight) irreducible maps
  6. Slice decomposition of maps
  7. Recursive decomposition of irreducible slices
  8. Recursive decomposition of a $\boldsymbol{2b}$-irreducible $\boldsymbol{2p}$-slice, case (I): $\boldsymbol{0\leq p\leq b-1}$.
  9. Recursive decomposition of a $\boldsymbol{2b}$-irreducible $\boldsymbol{2p}$-slice, case (II): $\boldsymbol{p= b}$.
  10. Recursive decomposition of a $\boldsymbol{2b}$-irreducible $\boldsymbol{2p}$-slice, case (III): the $\boldsymbol{2b}$-angle slice.
  11. Decorated tree formulation
  12. Recursive construction of the decorated tree, initialisation.
  13. Recursive construction of the decorated tree, case (I): $\boldsymbol{0\leq p\leq b-1}$.
  14. Recursive construction of the decorated tree, case (II): $\boldsymbol{p= b}$.
  15. Characterization of $b$-decorated trees.
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $n, b, m_1,\ldots,m_n$ be positive integers and let us denote by $\mathcal{N}_{n}^{(b)}(2m_1, \ldots,2m_n)$ the number of planar bipartite tight $2b$-irreducible maps with $n$ labeled faces of respective degrees $2m_1,\ldots,2m_n$. Then, for $n \geq 3$, $m_1 \geq b+1$ and $m_2,\ldots,m_n\geq b$, where $p^{(b)}_{k}(m)$ and $q^{(b)}_{k}(m)$ denote the polynomials in $b$ and $m$: and $\alpha^{(b

Figures (21)

  • Figure 1: A generic slice (left) and the two possible slices without inner faces: the trivial slice (middle) and the empty slice (right).
  • Figure 2: Sketch of the decomposition of a map $\mathbf{m}$ with two marked faces $1$ and $2$ (left) into elementary slices, by cutting its preimage $\tilde{\mathbf{m}}$ (right) along leftmost geodesics. In the notations of the main text, we have $m_1=6$, $m_2=4$ and $k=9$.
  • Figure 3: Examples of $2$-slices: a generic one (left)---note that it differs from the $0$-slice of Figure \ref{['fig:slice-types']} only by a shift of the base edge---and the $4$-angle slice (right). Both are $4$-irreducible.
  • Figure 4: Case (I) of the decomposition. When $p \leq b-1$ and the slice is not reduced to the $2b$-angle slice, we know that $A \neq C$ so we name $D_1$ the first vertex on the red boundary $P_0$. $P_1$ is the leftmost shortest path from C to A avoiding $CD_1$. Cutting along $P_1$ yields a $2p_1$-slice $\boldsymbol{\sigma}_1$ with apex $A_1$ and base edge $CD_1$. We start again in the rest of the slice (which is now a $2(p-p_1)$-slice with base $BC$), until $P_q$ goes through $BC$ and the blue boundary (here $q=3$).
  • Figure 5: Case (II) of the decomposition, here for $p=b=1$. This 2-irreducible 2-slice is composed of a face $f$ of degree $10$, and $9$ elementary slices. Left: we draw the (green) leftmost geodesic towards the apex from each vertex incident to $f$; this delimits the slices $\boldsymbol{\sigma}_1, \ldots, \boldsymbol{\sigma}_9$ (here, the slices $\boldsymbol{\sigma}_1, \boldsymbol{\sigma}_6$ and $\boldsymbol{\sigma}_8$ are empty, and $\boldsymbol{\sigma}_2, \boldsymbol{\sigma}_5$ and $\boldsymbol{\sigma}_9$ are trivial). The small labels in the corners of $f$ are the proximities to the apex, here equal to $5$ minus the length of a shortest path to $A$ avoiding $BC$: they go from $0$ at $B$ to $2b+1=3$ at $C$ by steps of $\pm 1$. Right: the result after cutting along the geodesics. The green arrows show how to glue back the elementary slice boundaries, in order to recover the original slice. Note that this slice is not tight, since it has a leaf $L$ incident to $f$.
  • ...and 16 more figures

Theorems & Definitions (50)

  • Theorem 1.1
  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • Proposition 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 40 more