Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We inv... more Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and determinant [Formula: see text] of odd degree [Formula: see text] having at least [Formula: see text] independent sections. This locus possesses a virtual fundamental class. We show that in many cases this class is nonzero, which implies that the Brill–Noether locus is nonempty. For many values of [Formula: see text] and [Formula: see text] the result is best possible. We obtain more precise results for [Formula: see text]. Appendix A contains the proof of a combinatorial lemma which we need.
Dedicated to Herb Wilf on the occasion of his 80th birthday Abstract. We define a de Bruijn proce... more Dedicated to Herb Wilf on the occasion of his 80th birthday Abstract. We define a de Bruijn process with parameters n and L as a certain continuous-time Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as vertices. We determine explicitly its steady state distribution and its characteristic polynomial, which turns out to decompose into linear factors. In addition, we examine the stationary state of two specializations in detail. In the first one, the de Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de Bruin process, the distribution has constant density but nontrivial correlation functions. The two point correlation function is determined using generating function techniques. 1.
Strehl, V., Identities of Rothes-Abel-Schläfli-Hurwitz-type, Discrete Mathematics 99 (1992) 32r-3... more Strehl, V., Identities of Rothes-Abel-Schläfli-Hurwitz-type, Discrete Mathematics 99 (1992) 32r-340. Several convolution identities, containing many free parameters, are shown to follow in a very simple way from a combinatorial construction. By specialization of the parameters one can find many of the known generalizations or variations of Abel's generalization of the binomial theorem, including those obtained by Rothe, Schläfli, and Hurwitz. A convolution identity related to Mellin's expansion of algebraic functions, proposed recently by Louck (but contained in equivalent form in earlier work by Raney and Mohanty), and a counting formula for labelled trees by rising edges, due to Kreweras, are also shown to follow from the general approach.
These comments contain a somewhat shorter proof of Atkinson's Theorem 2 and give some pointers to... more These comments contain a somewhat shorter proof of Atkinson's Theorem 2 and give some pointers to closely related literature. Let s_n denote the set of skew-merged permutations of [1::n]. The characteristic property of these permutations is precisely described in the title of Atkinson's article. Each such permutation, if represented as a cloud of points on a n n-grid in the traditional manner, has a number k (where 0 <= k <=n) of white elements (see Atkinson, Theorem 1). These are the elements that simultaneously belong to an increasing subsequence of maximum length and to a decreasing subsequence of maximum length. We will denote by t_k(n) the set elements of s_n with precisely k white elements (which form an increasing or decreasing sequence of contiguous elements). For further reference we introduce the set y_n: = t_1(n)+t_2(n)+:::+ t_n(n) of skew-merged permutations with at least one white element. In the sequel I use the symbols s_n, t_k(n),y_n also to denote the cardinalities...
Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We inv... more Let [Formula: see text] be a smooth projective complex curve of genus [Formula: see text]. We investigate the Brill–Noether locus consisting of stable bundles of rank 2 and determinant [Formula: see text] of odd degree [Formula: see text] having at least [Formula: see text] independent sections. This locus possesses a virtual fundamental class. We show that in many cases this class is nonzero, which implies that the Brill–Noether locus is nonempty. For many values of [Formula: see text] and [Formula: see text] the result is best possible. We obtain more precise results for [Formula: see text]. Appendix A contains the proof of a combinatorial lemma which we need.
Dedicated to Herb Wilf on the occasion of his 80th birthday Abstract. We define a de Bruijn proce... more Dedicated to Herb Wilf on the occasion of his 80th birthday Abstract. We define a de Bruijn process with parameters n and L as a certain continuous-time Markov chain on the de Bruijn graph with words of length L over an n-letter alphabet as vertices. We determine explicitly its steady state distribution and its characteristic polynomial, which turns out to decompose into linear factors. In addition, we examine the stationary state of two specializations in detail. In the first one, the de Bruijn-Bernoulli process, this is a product measure. In the second one, the Skin-deep de Bruin process, the distribution has constant density but nontrivial correlation functions. The two point correlation function is determined using generating function techniques. 1.
Strehl, V., Identities of Rothes-Abel-Schläfli-Hurwitz-type, Discrete Mathematics 99 (1992) 32r-3... more Strehl, V., Identities of Rothes-Abel-Schläfli-Hurwitz-type, Discrete Mathematics 99 (1992) 32r-340. Several convolution identities, containing many free parameters, are shown to follow in a very simple way from a combinatorial construction. By specialization of the parameters one can find many of the known generalizations or variations of Abel's generalization of the binomial theorem, including those obtained by Rothe, Schläfli, and Hurwitz. A convolution identity related to Mellin's expansion of algebraic functions, proposed recently by Louck (but contained in equivalent form in earlier work by Raney and Mohanty), and a counting formula for labelled trees by rising edges, due to Kreweras, are also shown to follow from the general approach.
These comments contain a somewhat shorter proof of Atkinson's Theorem 2 and give some pointers to... more These comments contain a somewhat shorter proof of Atkinson's Theorem 2 and give some pointers to closely related literature. Let s_n denote the set of skew-merged permutations of [1::n]. The characteristic property of these permutations is precisely described in the title of Atkinson's article. Each such permutation, if represented as a cloud of points on a n n-grid in the traditional manner, has a number k (where 0 <= k <=n) of white elements (see Atkinson, Theorem 1). These are the elements that simultaneously belong to an increasing subsequence of maximum length and to a decreasing subsequence of maximum length. We will denote by t_k(n) the set elements of s_n with precisely k white elements (which form an increasing or decreasing sequence of contiguous elements). For further reference we introduce the set y_n: = t_1(n)+t_2(n)+:::+ t_n(n) of skew-merged permutations with at least one white element. In the sequel I use the symbols s_n, t_k(n),y_n also to denote the cardinalities...
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