Annals of Combinatorics
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Dyson’s “Favorite” Identity and Chebyshev Polynomials of the Third and Fourth Kind
Annals of Combinatorics - Tập 23 - Trang 443-464 - 2019
The combinatorial and analytic properties of Dyson’s “favorite” identity are
studied in detail. In particular, a q-series analog of the anti-telescoping
method is used to provide a new proof of Dyson’s results with mock theta
functions popping up in intermediate steps. This leads to the appearance of
Chebyshev polynomials of the third and fourth kind in Bailey pairs related to
Bailey’s Lemma. The ...
Chain Decompositions of q, t-Catalan Numbers via Local Chains
Annals of Combinatorics - Tập 24 - Trang 739-765 - 2020
The q, t-Catalan number $${{\,\mathrm{Cat}\,}}_n(q,t)$$ enumerates integer
partitions contained in an $$n\times n$$ triangle by their dinv and external
area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018)
proposed a new approach to understanding the symmetry property
$${{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)$$ based on decomposing
the set of all integer par...
From Quantum Cohomology to Algebraic Combinatorics: The Example of Flag Manifolds
Annals of Combinatorics - Tập 4 - Trang 299-305 - 2000
The computation and understanding of quantum cohomology is a very hard problem
in mathematical physics (string theory). We review in non-technical terms how,
in the case of the flag manifolds, this problem can turn out to be at its core a
non-trivial problem in algebraic combinatorics.
The Asymptotic Behavior of Certain Birth Processes
Annals of Combinatorics - Tập 10 - Trang 255-269 - 2006
We describe a connection between discrete birth process and a certain family of
multivariate interpolation polynomials. This enables us to compute all
asymptotic moments of the birth process, generalizing previously known results
for the mean and variance.
Walks with Small Steps in the 4D-Orthant
Annals of Combinatorics - Tập 25 - Trang 153-166 - 2021
We provide some first experimental data about generating functions of restricted
lattice walks with small steps in $${\mathbb {N}}^4$$ .
Controllable Subsets in Graphs
Annals of Combinatorics - Tập 16 - Trang 733-744 - 2012
Let X be a graph on ν vertices with adjacency matrix A, and let S be a subset of
its vertices with characteristic vector z. We say that the pair (X, S) is
controllable if the vectors A r z for r = 1, . . . , ν − 1 span
$${\mathbb{R}^{\nu}}$$ . Our concern is chiefly with the cases where S = V(X),
or S is a single vertex. In this paper we develop the basic theory of
controllable pairs. We will see...
Các phân hoạch đồng dư tuần tự và các ánh xạ liên quan Dịch bởi AI
Annals of Combinatorics - - 2019
Chúng tôi nghiên cứu một lớp phân hoạch thú vị, các phần của nó tuân theo một
điều kiện đồng dư cực kỳ nghiêm ngặt mà chúng tôi gọi là "đồng dư tuần tự": phần
thứ m đồng dư với phần thứ $$(m+1)$$ theo modulo m, trong đó phần nhỏ nhất đồng
dư với không theo modulo độ dài của phân hoạch. Hóa ra những đối tượng có vẻ tối
tăm này được nhúng một cách tự nhiên trong lý thuyết phân hoạch. Chúng tôi cho
t...
#phân hoạch #đồng dư tuần tự #ánh xạ một-một #lý thuyết lý tưởng phân hoạch
Phylogenetic Invariants for $${\mathbb{Z}_3}$$ Scheme-Theoretically
Annals of Combinatorics - Tập 20 - Trang 549-568 - 2016
We study phylogenetic invariants of general group-based models of evolution with
group of symmetries $${\mathbb{Z}_3}$$ . We prove that complex projective
schemes corresponding to the ideal I of phylogenetic invariants of such a model
and to its subideal $${I'}$$ generated by elements of degree at most 3 are the
same. This is motivated by a conjecture of Sturmfels and Sullivant [14, Conj.
29], whi...
On t-Core Towers and t-Defects of Partitions
Annals of Combinatorics - Tập 21 - Trang 119-130 - 2017
We study generating functions which count the sizes of t-cores of partitions,
and, more generally, the sizes of higher rows in t-core towers. We then use
these results to derive an asymptotic results for the average size of the
t-defect of partitions, as well as some curious congruences.
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