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16
Walks with small steps in the quarter plane
 Contemporary Mathematics
"... Abstract. Let S ⊂ {−1, 0,1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a mo ..."
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Cited by 47 (7 self)
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Abstract. Let S ⊂ {−1, 0,1} 2 \ {(0, 0)}. We address the enumeration of plane lattice walks with steps in S, that start from (0, 0) and always remain in the first quadrant {(i, j) : i ≥ 0, j ≥ 0}. A priori, there are 2 8 problems of this type, but some are trivial. Some others are equivalent to a model of walks confined to a halfplane: such models can be solved systematically using the kernel method, which leads to algebraic generating functions. We focus on the remaining cases, and show that there are 79 inherently different problems to study. To each of them, we associate a group G of birational transformations. We show that this group is finite (of order at most 8) in 23 cases, and infinite in the 56 other cases. We present a unified way of solving 22 of the 23 models associated with a finite group. For all of them, the generating function is found to be Dfinite. The 23rd model, known as Gessel’s walks, has recently been proved by Bostan et al. to have an algebraic (and hence Dfinite) solution. We conjecture that the remaining 56 models, associated with an infinite group, have a nonDfinite generating function. Our approach allows us to recover and refine some known results, and also to obtain new
On kcrossings and knestings of permutations
 Proc. 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), DMTCS proc
, 2010
"... Abstract. We introduce kcrossings and knestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of knoncrossing permutations is equal to the number of knonnesting permutations. We also provid ..."
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Cited by 8 (3 self)
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Abstract. We introduce kcrossings and knestings of permutations. We show that the crossing number and the nesting number of permutations have a symmetric joint distribution. As a corollary, the number of knoncrossing permutations is equal to the number of knonnesting permutations. We also provide some enumerative results for knoncrossing permutations for some values of k. Résumé. Nous introduisons les kchevauchement d’arcs et les kempilements d’arcs de permutations. Nous montrons que l’index de chevauchement et l’index de empilement ont une distribution conjointe symétrique pour les permutations de taille n. Comme corollaire, nous obtenons que le nombre de permutations n’ayant pas un kchevauchement est égal au nombre de permutations n’ayant un kempilement. Nous fournissons également quelques résultats énumératifs.
A generating tree approach to knonnesting partitions and permutations
, 2011
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FAMILIES OF PRUDENT SELFAVOIDING WALKS
 JOURNAL OF COMBINATORIAL THEORY SERIES A 117, 3 (2010) 313344
, 2010
"... A selfavoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Préa i ..."
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Cited by 7 (1 self)
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A selfavoiding walk (SAW) on the square lattice is prudent if it never takes a step towards a vertex it has already visited. Prudent walks differ from most classes of SAW that have been counted so far in that they can wind around their starting point. Their enumeration was first addressed by Préa in 1997. He defined 4 classes of prudent walks, of increasing generality, and wrote a system of recurrence relations for each of them. However, these relations involve more and more parameters as the generality of the class increases. The first class actually consists of partially directed walks, and its generating function is wellknown to be rational. The second class was proved to have an algebraic (quadratic) generating function by Duchi (2005). Here, we solve exactly the third class, which turns out to be much more complex: its generating function is not algebraic, nor even Dfinite. The fourth class — general prudent walks — is the only isotropic one, and still defeats us. However, we design an isotropic family of prudent walks on the triangular lattice, which we count exactly. Again, the generating function is proved to be nonDfinite. We also study the asymptotic properties of these classes of walks, with the (somewhat disappointing) conclusion that their endpoint moves away from the origin at a positive speed. This is confirmed visually by the random generation procedures we have designed.
Enumeration of bilaterally symmetric 3noncrossing partitions
 Discrete Math
"... Abstract. Schützenberger’s theorem for the ordinary RSK correspondence naturally extends to Chen et. al’s correspondence for matchings and partitions. Thus the counting of bilaterally symmetric knoncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous resul ..."
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Cited by 4 (2 self)
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Abstract. Schützenberger’s theorem for the ordinary RSK correspondence naturally extends to Chen et. al’s correspondence for matchings and partitions. Thus the counting of bilaterally symmetric knoncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3noncrossing partitions, we use a different technique to develop a MAPLE package for 2dimensional vacillating lattice walk enumeration problems. As an application, we find an interesting relation between two special bilaterally symmetric partitions.
THE EXPECTED NUMBER OF INVERSIONS AFTER n ADJACENT TRANSPOSITIONS
, 2010
"... We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group Sm+1. We then derive from this expression the asymptotic behaviour of this number when n ≡ nm scales with m in various ways. Our starting point is an equivalence ..."
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Cited by 2 (0 self)
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We give a new expression for the expected number of inversions in the product of n random adjacent transpositions in the symmetric group Sm+1. We then derive from this expression the asymptotic behaviour of this number when n ≡ nm scales with m in various ways. Our starting point is an equivalence, due to Eriksson et al., with a problem of weighted walks confined to a triangular area of the plane.
Increasing and decreasing sequences of length two in 01fillings of moon polyominoes
, 2008
"... We put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings and set partitions in the larger context of the enumeration of increasing and decreasing sequences of length 2 in fillings of moon polyominoes. ..."
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Cited by 1 (1 self)
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We put recent results on the symmetry of the joint distribution of the numbers of crossings and nestings of two edges over matchings and set partitions in the larger context of the enumeration of increasing and decreasing sequences of length 2 in fillings of moon polyominoes.