... .
the electronic journal of combinatorics 10 (2003), #R3 5
Proof. The proof is by induction on the number of vertices of a graph. The n -cube Q
n
,
n ≥ 1, can be obtained as the expansion of the ... graphs U
e
and derive
c
(s+1)
(G, −1) =
e∈F(G)
θ
s
(U
e
)=θ
s+1
(G).
the electronic journal of combinatorics 10 (2003), #R3 8
The cube polynomial and its de...
... j)
1
=
(3n + j −1)(n − j)
(2j + 1)(2j −1)
.
Theorem 2. The expectation and the variance of the parameter number of descendants
of node j in a random heap ordered tree of size n is given by
Expectation ... together
with a bijection from the nodes to the set {1, ,n} which is monotonically increasing
when going from the root to the leaves. We consider the number of de...
... gives the position of the first occurrence
of green on the right part of the X-axis, h the number of the first line, above the X-axis,
where green appears, and α the position on this line of the ... a subset S
R
of R
and an integer α with 0 ≤ α<w;thesetS obtained as the union of the classes of the
elements of S
R
in the tiling induced by the...
... 1, in the expansion of
κ(ω). To distinguish the steps of the original path from the steps of the inserted paths
we will colour the former red and the latter blue. This leads to the idea of bi-coloured
Dyck ... partition the
path into y + 1 subpaths. The first of these is the Dyck path β
0
and the others are of the
form u
α
β
α
, α =1, y,whereu
α
is a t...
...
electrocardiographic findings and the relief of his chest pain after stopping the cayenne
pepper pills support the role of capsaicin in the development of coronary vasospasm
through increased sympathetic stimulation. ...
Authors’ contributions
OS and YS treated the patient and wrote the case report. HK and MTG critically revised
the manuscript. All authors...
... Labels and Symmetry Numbers
Let T be an element of C,andletv be a vertex of T . We define the symmetry
label l
T
(v)ofv as follows: If v is the root of T ,thenl
T
(v)=1. Ifv is not the
root of T, then ... largest in their equivalence class as
standard trees, and use them as equivalence class representatives. We will de-
note the set of standard Catalan trees by standard(C...
... statements of the theorem are implied by Theorem 1 and the following lemma.
Lemma 1. If a is a member of the Catalan family then a = δa. Otherwise, a<δa.
Proof. The proof is by induction on the generation ... direct proof of
Theorem 3 and the reason is that there is an important difference in the distribution of
labels in the Catalan family tree and the tree...
... each, and other random variables.
Equiprobable allocations are the most simple and well studied. Consider, for example,
an Internet voting on the theme: “Which of the N teams will win the world ... words with the same
hash value.
The model of random equiprobable allocations of particles (hash values of different
input words) into N cells (elements of B) is often used...
... remove the edges of all white a
i
and black b
i
vertices on one side of ,
and the edges of all black a
i
and white b
i
vertices on the other side of .LetG
+
be the
subgraph on one side of ,andG
−
the ... n)=2
n−m
(2k
m;n
+1)
2
.
Proof. The proof is analogous to the proof of the previous corollary, and once again the
odd factor 2k
m;n
+ 1 is the n...