Andrews, Hirschhorn, and Sellers studied the partition function
ped
(
n
) which enumerates the number of partitions of
n
with even parts distinct, and obtained a number of interesting congruences. ...This paper aims to introduce a partition statistic to investigate the partition function
ped
(
n
). We give combinatorial interpretations for some properties of
ped
(
n
) including the infinite families of congruences given by Andrews et al.
Compacted binary trees admit a stretched exponential Elvey Price, Andrew; Fang, Wenjie; Wallner, Michael
Journal of combinatorial theory. Series A,
January 2021, 2021-01-00, 2021-01, Letnik:
177
Journal Article
Recenzirano
Odprti dostop
A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of ...compacted binary trees of size n grows asymptotically likeΘ(n!4ne3a1n1/3n3/4), where a1≈−2.338 is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm of quadratic arithmetic complexity for computing the number of compacted trees up to a given size. We use empirical methods to estimate the values of all terms defined by the recurrence, then we prove by induction that these estimates are sufficiently accurate for large n to determine the asymptotic form. Our results also lead to new bounds on the number of minimal finite automata recognizing a finite language on a binary alphabet. As a consequence, these also exhibit a stretched exponential.
Bargraphs are a special class of column-convex polyominoes. They can be identified with lattice paths with unit steps north, east, and south that start at the origin, end on the x-axis, and stay ...strictly above the x-axis everywhere except at the endpoints. Bargraphs, which are used to represent histograms and to model polymers in statistical physics, have been enumerated in the literature by semiperimeter and by several other statistics, using different methods such as the wasp-waist decomposition of Feretić, and a bijection with certain Motzkin paths.
In this paper we describe an unusual bijection between bargraphs and Dyck paths, and study how some statistics are mapped by the bijection. As a consequence, we obtain a new interpretation of Catalan numbers, as counting bargraphs where the semiperimeter minus the number of peaks is fixed.
Given a real reductive linear Lie group
G
R
, the Mackey analogy is a bijection between the set of irreducible tempered representations of
G
R
and the set of irreducible unitary representations of ...its Cartan motion group, established by Higson and Afgoustidis. We show that this bijection arises naturally from families of twisted
D
-modules over the ag variety of
G
R
.
A hybrid binary tree is a complete binary tree where each internal node is labeled with 1 or 2, but with no left (1, 1)-edges. In this paper, we consider enumeration of the set of hybrid binary trees ...according to the number of internal nodes and some other combinatorial parameters. We present enumerative results by giving Riordan arrays, bivariate generating functions, as well as closed formulas. As a consequence, we obtain some new combinatorial matrices, one of which is analogous to the Borel triangle. We also present a bijection between the set of all hybrid binary trees with
n
internal nodes and the set of generalized Schröder paths from (0, 0) to (2
n
, 0) which are consist of up steps
u
=
(
1
,
1
)
, horizontal steps
h
=
(
2
,
0
)
, down steps
d
=
(
1
,
-
1
)
, and double up steps
U
=
(
2
,
2
)
, and never travel below the
x
-axis.
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