Báo cáo toán học: "On Han’s Hook Length Formulas for Trees" pdf
Báo cáo toán học: "On Han’s Hook Length Formulas for Trees" pdf
... Classifications: 05A15, 05A19
Abstract
Recently, Han obtained two hook length formulas for binary trees and asked
for combinatorial proofs. One of Han’s formulas has been generalized to k-ary
trees by Yang. ... staircase labeling.
1 Introduction
Motivated by the hook length formula of Postnikov [6], Han [4] discovered two hook length
formulas for binary trees. Han’s...
Báo cáo toán học: "On Stanley’s Partition Function" doc
... the
number of hooks of even length. For the definition of hook lengths, see Stanley [7, p.
373]. A hook of even length is called an even hook. The following theorem shows that
the number of even hooks ... Furthermore, we derive formulas for the generating functions for the numbers
u(4n), u(4n + 1), u(4n + 2) and u(4n + 3), which are analogous to the generating function
formula...
Báo cáo toán học: "On Kadell’s two Conjectures for the q-Dyson Product" pot
... =
n
k=0
N(k, I) − N(k, J)
w
k
(T ). (2.4)
We need the explicit formula for the case i
1
= 0 for our calculation. As stated in [8],
the formula for this case can be derived using an action π on Laurent ... q-Dyson
product, and then applying the formulas of [8]. Unfortunately, we are not able to derive
a nice for mula.
Our second objective is to contribute a q-analogous formula...
Báo cáo toán học: "On Rowland’s sequence Fernando Chamizo, Dulcinea Raboso and Seraf" ppsx
... 1.2 For any sequence of the form (2), there exists a positive integer N such
that a
k
− a
k−1
∈ P
1
for every k > N.
Actually in [4] this conjecture is stated for starting values of the form ... for the first time for k = 66, which corresponds to c
4
= 2r
4
− 1. But
neither r
4
= 66 nor c
3
= 99 are prime.
2 Relation between the conjectures
We start by giving the alternative formu...
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... M
n
(x; a, b, c) for root system of type BC by using two well-known results. One result
is Jacobi’s change of variable formula. See, e.g., [9].
Theorem 11 (Jacobi’s Residue Formula). Let y = ... a Laurent series and
let b be the integer such that f(x)/x
b
is a formal power series with nonzero constant term.
Then for any formal series G( y) such that the composition G(f(x)) is a Laurent .....
Báo cáo toán học: " On the Laws of Large Numbers for Blockwise Martingale Differences and Blockwise Adapted Sequences" potx
... remain satisfied for the sequences consisting of
independent blocks. However, the same problem for sequences of blockwise in-
dependent and identically distributed random variables and for blockwise ... q>1,then
lim
n→∞
q
−n
n
k=0
q
k+1
x
k
=0.
Proof. Let s = q +
∞
i=0
q
−i
. For any >0, there exists k
0
such that |x
k
| <
2s
for all k ≥ k
0
. Since lim
n→∞
q
−n
=0,so...
Báo cáo toán học: "On the distribution of depths in increasing trees" ppsx
... general, there is no explicit expression for the
integral, but for very simple families of increasing trees, the formula simplifies:
• If Φ(t) = e
ct
for some c > 0, then T(z, 1) =
1
c
log
1
1−cz
, ... consequence. For very simple families of increasing trees, these theorems occur in the
aforementioned pap er by Dobrow and Smythe [7]. Our bijection provides a simple combi-
natorial...
Báo cáo toán học: "On multicolor Ramsey number of paths versus cycles" pdf
... P
4
, C
k
) = k + 2 for k ≥ 11,
(ii) ([4]) R(P
3
, P
4
, C
k
) = k + 1 for k ≥ 12,
(iii) ([13]) R(P
4
, P
5
, C
k
) = k + 2 for k ≥ 23,
(iv) ([13]) R(P
4
, P
6
, C
k
) = k + 3 for k ≥ 18.
We end ... integer. Then
(i) R(P
3
, P
6
, C
k
) = k + 2 for k ≥ 23,
(ii) R(P
6
, P
6
, C
k
) = R(P
4
, P
8
, C
k
) = k + 4 for k ≥ 27,
(iii) R(P
6
, P
7
, C
k
) = k + 4 for k ≥ 57.
3 Some thre...
Báo cáo toán học: "The Prime Power Conjecture is True for n pdf
... toatranslateg + D of
itself for some g ∈ G.Ifα is of the form α : x → tx for t ∈ relatively prime to
the order of G,thenα is called a numerical multiplier. Most nonexistence results
for difference sets ... difference sets of
order n exist only for n a prime power. Evans and Mann [2] verified this for cyclic
difference sets for n ≤ 1600. In this paper we verify the PPC for n ≤ 2,000,...
Báo cáo toán học: "Hook Length Formulas for Trees by Han’s Expansion" doc
... section, we derive hook length expansion formulas for plane trees and plane forests,
and we find certain g enerating functions which lead to several hook length formulas. Some
known formulas can be ... 11
we consider the expansion formulas for plane trees and plane forests. In Section 4, we
present expansion formulas and hook length formulas for labeled trees and f...