... =1,wehavethatp
1
+ αc
1
≤ [x]e
C(x)
= c
1
and so c
1
≤−p
1
/(α − 1) = d
1
. Hence the claim is true when n = 1. Suppose the
claim is true for all values less than n.Wehave
p
n
+ αc
n
≤ [x
n
]e
C(x)
=[x
n
]exp(c
1
x ... following:
1. C(x), H(x), and A(x) all have zero radius of convergence;
2. c
n
,a
n
, and n!h
n
are positive integers;
3. A(x)=exp(H(x)) = exp
j≥1
C(x
j
)/j
;
4. lim...
... parity-reversing.
It follows that in the signed sum of walks in W all terms corresponding to walks in N
cancel, leaving only the walks that correspond to permutations of {1, 2, ,n/2} with no
ascending ... correspondence with lattice walks. Interestingly, however, by
providing a combinatorial proof of this correspondence, in section 3 below, we will be giving
an indepe...
... obtain
1
N
(ξ)=
d
j=1
sin πξ
j
πξ
j
− F(ξ)
d
j=1
sin πδ
j
ξ
j
πξ
j
, (11)
lattice tilings by cubes: whole,
notched and extended
Mihail Kolountzakis
Department of Mathematics
1409 W. Green St.,
University ... (11)). We find all
the tilings discovered by Stein, which, by a deeper theorem of Schmerl [Sch], is the
complete list of possible translational tilings (lattice...
... lattice paths, d[m, n]=d[m−1,n]+d[m, n − 1] + γd[m − 1,n−1]. The
paths take horizontal, vertical, and diagonal steps (a king). If we write = à − 1,
then this recurrence enumerates lattice paths ... Brownian motion between parallel
reflecting walls, a continuous version equivalent to the asymptotic distribution of the
Kolmogorov-Smirnov test. The exact number of ordinary paths betwe...
... and
that there is a natural bijection between these permutations and lattice paths from
(0, 0) to (j, j) using steps (0, 1) and (1, 0) which never go above the main diagonal.
Also, let C(x)=
n≥0
c
n
x
n
=
1−
√
1−4x
2x
. ... diagonal (say it
stays below). Suppose it touches the diagonal first at the point (i, i). If there is
an entry of rank more than 1 inserted between the ith an...
... orbits. In the present case a family
is completely specified by the integer q ≡ q
1
which counts the traversals of the loop 1,
i.e., the number of letters 1 in the code word. Each of these q letters ... orthogonal polynomials of integers x with 0 ≤ x ≤ N.Theyhave
quite diverse applications ranging from the theory of covering codes [6] to the statistical
mechanics...
... the lattice structure comes from orientations of the completion.
2 Lattices of Fixed Degree Orientations
A plane graph is a planar graph G =(V, E) together with a fixed planar embedding.
In particular ... there may exist rigid
the electronic journal of combinatorics 11 (2004), #R15 15
Lattice Structures from Planar Graphs
Stefan Felsner
Technische Universităat Berlin, Institut fău...
... property,orisaHPP matroid.
the electronic journal of combinatorics 11(2) (2005), #A1 3
Matroid inequalities from electrical network theory
David G. Wagner
∗
Department of Combinatorics and Optimization
University ... 1981, Stanley applied the Aleksandrov–Fenchel Inequalities to prove a loga-
rithmic concavity theorem for regular matroids. Using ideas from electrical network
t...
... of orbits of PGL(2,q). There are some known
results on 3-designs from PGL(2,q) in the literature, see for example [1, 4, 6]. In this
paper, we first determine the sizes of orbits from the actions ... in the above formula.
7 Orbit sizes and 3-designs from PGL(2,q)
We use the results of the previous sections to show the existence of a large number of new
3-designs. First we state the...
... ∈ M
k
(p)andq is obtained from p by deleting
i − 2 entries from 1, ,i− 1 and not deleting i},
B = {q | q ∈ M
k
(p)andq is obtained from p by deleting
i − 2 entries from 1, ,i− 1andi},
C = {q ... paper develops in three main directions – permutation reconstruction from multisets
and sets of minors as well as permutation reconstruction from sets of minors within certain
classes...
... surfaces. Interdiscip. Inf. Sci., 12(2):93–107, 2006.
the electronic journal of combinatorics 15 (2008), #N11 5
Lattice points in Minkowski sums
Christian Haase, Benjamin Nill, Andreas Paffenholz
∗
Institut ... 14M25
Abstract
Fakhruddin has proved that for two lattice polygons P and Q any lattice point
in their Minkowski sum can be written as a sum of a lattice point in P and one...
... n-cycles or neither is an n -cycle. We consider these
two cases separately.
Case 1. Suppose neither p nor q is an n -cycle. Let k be the length of the cycle in p (and
hence the length of the cycle ... permutation p “contains” the k -cycle (a
1
a
2
. . . a
k
), or the k -cycle is
“in” p, if it appears in the decomposition of p into disjoint cycles.
2 Reconstruction from Cycle Minors...
... and at least one of them contains an
end of an edge of A
i
∩ B
since at least one of G[X
i
∩ Y ] and G[X
i
∩ Y ] contains an end
of an edge of A
i
∩ B.
the electronic journal of combinatorics ... 05C10
Abstract
A new characterisation of planar graphs is presented. It concerns the structure
of the cocycle space of a graph, and is motivated by consideration of the...
... the second author
introduced and studied (δ, χ)-bounded families of graphs (under the name of δ-bounded
families) in [10]. The so-called color-bound family of graphs mentioned in the related
∗
Research ... orb(C) is (δ, χ)-bounded we may restrict
ourselves to bipartite graphs. We shall make use of this result in proving the following
theorems.
Similar to the concept of...