2,812 research outputs found
Simsun permutations, simsun successions and simsun patterns
In this paper, we introduce the definitions of simsun succession, simsun
cycle succession and simsun pattern. In particular, the ordinary simsun
permutations are permutations avoiding simsun pattern 321. We study the descent
and peak statistics on permutations avoiding simsun successions. We give a
combinatorial interpretation of the q-Eulerian polynomials introduced by Brenti
(J. Combin. Theory Ser. A 91 (2000), 137-170). We also present a bijection
between permutations avoiding simsun pattern 132 and set partitions.Comment: 11 page
The peak statistics on simsun permutations
In this paper, we study the relationship among left peaks, interior peaks and
up-down runs of simsun permutations. Properties of the generating polynomials,
including the recurrence relation, generating function and real-rootedness are
studied. Moreover, we introduce and study simsun permutations of the second
kind.Comment: 13 page
Learning Image Conditioned Label Space for Multilabel Classification
This work addresses the task of multilabel image classification. Inspired by
the great success from deep convolutional neural networks (CNNs) for
single-label visual-semantic embedding, we exploit extending these models for
multilabel images. Specifically, we propose an image-dependent ranking model,
which returns a ranked list of labels according to its relevance to the input
image. In contrast to conventional CNN models that learn an image
representation (i.e. the image embedding vector), the developed model learns a
mapping (i.e. a transformation matrix) from an image in an attempt to
differentiate between its relevant and irrelevant labels. Despite the
conceptual simplicity of our approach, experimental results on a public
benchmark dataset demonstrate that the proposed model achieves state-of-the-art
performance while using fewer training images than other multilabel
classification methods
Eulerian polynomials, perfect matchings and Stirling permutations of the second kind
In this paper, we first present combinatorial proofs of a kind of expansions
of the Eulerian polynomials of types A and B, and then we introduce Stirling
permutations of the second kind. In particular, we count Stirling permutations
of the second kind by their cycle ascent plateaus, fixed points and cycles.Comment: 16 page
Derivative polynomials and enumeration of permutations by their alternating descents
In this paper we present an explicit formula for the number of permutations
with a given number of alternating descents. Moreover, we study the interlacing
property of the real parts of the zeros of the generating polynomials of these
numbers.Comment: 6 page
Stirling permutations, cycle structures of permutations and perfect matchings
In this paper we provide a unified combinatorial approach to establish a
connection between Stirling permutations, cycle structures of permutations and
perfect matchings. The main tool of our investigations is MY-sequences. In
particular, we discover that the Eulerian polynomials have a simple
combinatorial interpretation in terms of some statistics on MY-sequences.Comment: 9 page
Gamma-positivity and partial gamma-positivity of descent-type polynomials
In this paper, we study gamma-positivity of descent-type polynomials by
introducing the change of context-free grammars method. We first present
grammatical proofs of the gamma-positivity of the Eulerian polynomials, type B
Eulerian polynomials, derangement polynomials, Narayana polynomials and type B
Narayana polynomials. We then provide partial gamma-positive expansions for
several multivariate polynomials associated to Stirling permutations,
Legendre-Stirling permutations, Jacobi-Stirling permutations and type B
derangements, and the recurrences for the partial gamma-coefficients of these
expansions are also obtained. Moreover, we define variants of the Foata-Strehl
group action which are used to give combinatorial interpretations for the
coefficients of most of these partial gamma-positive expansions.Comment: 31 page
The alternating run polynomials of permutations
In this paper, we first consider a generalization of the David-Barton
identity which relate the alternating run polynomials to Eulerian polynomials.
By using context-free grammars, we then present a combinatorial interpretation
of a family of q-alternating run polynomials. Furthermore, we introduce the
definition of semi-gamma-positive polynomial and we show the
semi-gamma-positivity of the alternating run polynomials of dual Stirling
permutations. A connection between the up-down run polynomials of permutations
and the alternating run polynomials of dual Stirling permutations is
established.Comment: 14 page
Recurrence relations for binomial-Eulerian polynomials
Binomial-Eulerian polynomials were introduced by Postnikov, Reiner and
Williams. In this paper, properties of the binomial-Eulerian polynomials,
including recurrence relations and generating functions are studied. We present
three constructive proofs of the recurrence relations for binomial-Eulerian
polynomials. Moreover, we give a combinatorial interpretation of the Betti
number of the complement of the k-equal real hyperplane arrangement.Comment: 14 page
Several variants of the Dumont differential system and permutation statistics
The Dumont differential system on the Jacobi elliptic functions was
introduced by Dumont (Math Comp, 1979, 33: 1293--1297) and was extensively
studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present
a labeling scheme for the cycle structure of permutations. We then introduce
two types of Jacobi-pairs of differential equations. We present a general
method to derive the solutions of these differential equations. As
applications, we present some characterizations for several permutation
statistics.Comment: 19 pages, to appear in Science China Mathematic
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