... the union of any element of F
1
with any element of
F
2
.
It is easy to see that the product of two cancellative families is also a cancellative
family ( (A
1
,A
2
) ∪(B
1
,B
2
)= (A
1
,A
2
) ∪(C
1
,C
2
) ... ask how large cancellative families can be. We define f(n) to be the size of the largest
possible cancellative family of subsets of a n-set and f(k,n)...
... 1996
Abstract
A method for constructing cubic graphs with girths in the range 13 to 16 is described. The
method is used to construct the smallest known cubic graphs for girths 14, 15 and 16.
Introduction
We ... JournalofGraphTheory,6, 1982, 1-22.
A Simple Method for Constructing Small Cubic Graphs of Girths
14, 15, and 16
Geoffrey Exoo
Departm...
... A simple algorithm for constructing
Szemer´edi’s Regularity Partition
Alan Frieze
Department of Mathematical Sciences,
Carnegie ... combinatorics 6 (1999), #R17 3
A partition satisfying the second criterion is called equitable. V
0
is called the
exceptional class.
Following [9], for every equitable partition P into k +1classes ... (ˆx
+
)
T
Wˆy
−
≥ ˆγ
√
pq/4. (The proof
for...
... survey of frequency squares.
A set of (n − 1)
2
/(m − 1) mutually orthogonal frequency squares (MOFS) of type
F (n; λ)isacomplete set of type F (n; λ).
We shall describe a construction for complete ... 2000.
Abstract
A construction is described for combining affine designs with complete sets of
mutually orthogonal frequency squares to produce other complet...
... referee for numerous comments and
suggestions which have led to a substantially improved paper.
An Asymptotic Expansion for the Number of
Permutations with a Certain Number of Inversions
Lane Clark
Department ... Eds., Handbook of Mathematical Functions with
Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1966.
[2] E .A. Bender, Central...
... v(G) denote the number of vertices, and (G) denote the number of edges. In this
paper, we construct a graph G having no two cycles with the same length which leads to
the following result.
Theorem. ... Chunhui Lai, On the size of graphs with all cycle having distinct length, Discrete
Math. 122(1993) 363-364.
[6] Chunhui Lai, The edges in a graph...
... is the number of times the integer i occurs in the sequence T . Note that the
sum of all b(i), 1 ≤ i ≤ n,isequaltol, the length of T .
3 A simple proof for the existence of exponentially
balanced ... two, or are two successive powers of two. A proof for the
existence of exponentially balanced Gray codes is derived. The proof is much sim-
p...
... 1986.
the electronic journal of combinatorics 12 (2005), #R18 8
Theorem 2.3 (Aztec diamond theorem) The number of domino tilings of the Aztec
diamond of order n is 2
n(n+1)/2
.
Remark: The proof of ... between domino tilings of an Aztec diamond and non-
intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by
means of...
... agree above row r and to the left of column c (and the
definitions of r and c) we see that the only possibility for /à is (r, c), completing the
proof of this case and the theorem.
Acknowledgements. ... 308, 9 (2008), 1524–1530.
the electronic journal of combinatorics 15 (2008), #N23 4
A sharp bound for the reconstruction of partitions
Vincent Vatter
Departme...
... interpretation of the closed formula (10) for the number of con-
vex permutominoes. We remark that the expression of f
n
resembles the expression
(1) for the number c
n
of convex polyominoes of semi-perimeter ... : the uppermost cell of the rightmost column of the polyomino has the maximal
ordinate among all the cells of the polyomino;
U2 :...
... that those formulae yield interesting power series expansions
for the logarithm function. This generalizes an infinite series of Lehmer for the
natural logarithm of 4.
1 The abelian case: the Mahler ... fundamental group of the knot complement. For a square matrix
M with entries in CG the trace tr
CG
(M) denotes the coefficient of the unit element in
the sum of the...
... b
i
in P
[k]
j,i
.
the electronic journal of combinatorics 16 (2009), #R70 18
A normalization formula for the Jack polynomials in
superspace and an identity on partitions
Luc Lapointe
∗
Instituto ... where σ
j
and ∂
j
are respectively the interchang e and divided difference operators
on the variables a
j
and a
j+1
(see for instance [6]). The main st...
... Beissinger, who was the first to explore [1]
combinatorial proofs using colored permutations, for their assistance.
References
[1] Janet Beissinger. Colorful proofs of generating function identities. ... follows for (π, f ) ∈ S
n,A
: If R
(π,f)
= ∅, let φ((π, f)) = (π, f).
Otherwise, let (i, j) ∈ R
(π,f)
be minimal under the lexicographic ordering of R
(π,f)
. Let
˜π = (i, j)...
... A recurrence relation for the “inv” analogue of
q-Eulerian polynomials
Chak-On Chow
Department of Mathematics and Information Technology,
Hong Kong Institute of Education,
10 Lo ... relation algebraically. In the final section, we give
a combinatorial proof of this recurrence relation.
2 The recurrence relation
We derive in the present section the rec...
... a short proof for the result of Noonan [3] that the number
of permutations of length n containing exactly one occurence of pattern 321 is
3
n
2n
n−3
.
(To be precise, Noonan considered permutations ... A short proof for the number of permutations
containing pattern 321 exactly once
Alexander Burstein
Department of Mathematics
Howard...