1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
1 | 1 | 1 | ||||||
2 | 0 | 1 | 1 | |||||
3 | 0 | 1 | 1 | 2 | ||||
4 | 0 | 1 | 2 | 2 | 5 | |||
5 | 0 | 2 | 4 | 5 | 5 | 16 | ||
6 | 0 | 5 | 10 | 14 | 16 | 16 | 61 | |
7 | 0 | 16 | 32 | 46 | 56 | 61 | 61 | 271 |
Citation: Valeria Chiado Piat, Sergey S. Nazarov, Andrey Piatnitski. Steklov problems in perforated domains with a coefficient of indefinite sign[J]. Networks and Heterogeneous Media, 2012, 7(1): 151-178. doi: 10.3934/nhm.2012.7.151
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The Euler numbers
$ 1 + \sum\limits_{n\ge 1} E_n \frac{x^n}{n!} = \tan x + \sec x. $ |
This is the sequence A000111 in [20]. In 1877 Seidel [19] defined the triangular array
$ En,k=En,k−1+En−1,n+1−k(n≥k≥2) $
|
(1) |
with
$
E1,1E2,1→E2,2E3,3←E3,2←E3,1E4,1→E4,2→E4,3→E4,4⋯ = 10→11←1←00→1→2→2⋯
$
|
(2) |
The first few values of
1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
1 | 1 | 1 | ||||||
2 | 0 | 1 | 1 | |||||
3 | 0 | 1 | 1 | 2 | ||||
4 | 0 | 1 | 2 | 2 | 5 | |||
5 | 0 | 2 | 4 | 5 | 5 | 16 | ||
6 | 0 | 5 | 10 | 14 | 16 | 16 | 61 | |
7 | 0 | 16 | 32 | 46 | 56 | 61 | 61 | 271 |
André [1] showed in 1879 that the Euler number
$ {\mathcal{DU} }_4 = \{2143, 3142, 3241, 4132, 4231\}. $ |
In 1933 Kempener [14] used the boustrophedon algorithm (2) to enumerate alternating permutations without refering to Euler numbers. Since Entringer [7] first found the combinatorial interpretation of Kempener's table
Theorem 1 (Entringer). The number of the (down-up) alternating permutations of
$ {{\mathcal{DU} }_{n,k}} : = \left\{{\sigma\in{{\mathcal{DU} }_n} : \sigma_1 = k}\right\}. $ |
According to Foata-Schützenberger [9] a sequence of sets
The Springer numbers
$ 1 + \sum\limits_{n\ge 1} S_n \frac{x^n}{n!} = \frac{1}{\cos x - \sin x}. $ |
Arnold [2,p.11] showed in 1992 that
$ 1\,\bar{2}\,3, \; 1\,\bar{3}\,2, \; 1\,\bar{3}\,\bar{2}, \; 2\,1\,3, \; 2\,\bar{1}\,3, \; 2\,\bar{3}\,1, \; 2\,\bar{3}\,\bar{1}, \; 3\,1\,2, \; 3\,\bar{1}\,2, \; 3\,\bar{2}\,1, \; 3\,\bar{2}\,\bar{1}, $ |
where we write
$
S1,−1S2,2←S2,1S3,−3→S3,−2→S3,−1S4,4←S4,3←S4,2←S4,1⋯⇕12←10→2→316←16←14←11⋯ \;\;\;S1,1S2,−1←S2,−2S3,1→S3,2→S3,3S4,−1←S4,−2←S4,−3←S4,−4⋯⇕11←03→4→411←8←4←0⋯
$
|
where
$ Sn,k={Sn,k−1+Sn−1,−k+1if n≥k>1,Sn,−1if n>k=1,Sn,k−1+Sn−1,−kif −1≥k>−n. $
|
(3) |
Theorem 2 (Arnold). For all integers
$ {{\mathcal{S}}_{n,k}} : = \left\{{\sigma\in{{\mathcal{S}}_n} : \sigma_1 = k}\right\}. $ |
Moreover, for all integers
$ Sn,k=#{σ∈DUn(B):σ1=k}. $
|
Similarly, the numbers
-6 | -5 | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 | 5 | 6 | ||
1 | 1 | 1 | |||||||||||
2 | 0 | 1 | 1 | 2 | |||||||||
3 | 0 | 2 | 3 | 3 | 4 | 4 | |||||||
4 | 0 | 4 | 8 | 11 | 11 | 14 | 16 | 16 | |||||
5 | 0 | 16 | 32 | 46 | 57 | 57 | 68 | 76 | 80 | 80 | |||
6 | 0 | 80 | 160 | 236 | 304 | 361 | 361 | 418 | 464 | 496 | 512 | 512 |
This paper is organized as follows. In Section 2, we shall give the necessary definitions and present our main results. The proof of our theorems will be given in Sections 3-4. In Section 5, we shall give more insightful description of two important bijections. More precisely, Chuang et al.'s constructed a
Let
For each vertex
Definition 3. Given an increasing 1-2 tree
Let
$ {\mathcal{T}}_{n,k} = \left\{{T\in{\mathcal{T}}_n : {\rm{Leaf}}(T) = k}\right\}. $ |
Donaghey [5] (see also [3]) proved bijectively that the Euler number
Theorem 4 (Gelineau-Shin-Zeng). There is an explicit bijection
$ {\rm{Leaf}}(\psi(\sigma)) = {\rm{First}}(\sigma) $ |
for all
Let
Hetyei [12,Definition 4] defined recursively André permutation of second kind if it is empty or satisfies the following:
(ⅰ)
(ⅱ)
(ⅲ) For all
It is known that the above definition for André permutation of second kind is simply equivalent to the following definition. Let
Definition 5. A permutation
For example, the permutation
$ τ[1]=1,τ[2]=12,τ[3]=312,τ[4]=3124,τ[5]=31245. $
|
Foata and Schützenberger [10] proved that the Euler number
$ {\mathcal{A}}_{4} = \left\{{1234, 1423, 3124, 3412, 4123}\right\}. $ |
Remark. Foata and Schützenberger in [10] introduced augmented André permutation is a permutation
$ σj−1=max{σj−1,σj,σk−1,σk}andσk=min{σj−1,σj,σk−1,σk}, $
|
there exists
Definition 6. A permutation
By definition, an André permutations is always a Simsun permutation, but the reverse is not true. For example, the permutation
$ τ[1]=1,τ[2]=21,τ[3]=213,τ[4]=2134,τ[5]=25134. $
|
Let
$ {\mathcal{RS}}_{3} = \left\{{123,132,213,231,312}\right\}. $ |
As for
$ An,k:={σ∈An:σn=k},RSn,k:={σ∈RSn:σn=k}. $
|
Some examples are shown in Table 3.
Foata and Han [8,Theorem 1 (ⅲ)] proved that
Theorem 7. For positive integer
$ Leaf(T)=Last(ω(T)) $
|
(4) |
for all
Whereas one can easily show that the cardinality
Stanley [22,Conjecture 3.1] conjectured a refinement of Purtill's result [18,Theorem 6.1] about the
Theorem 8 (Hetyei).
For all
$ #An,k=#RSn−1,k−1. $
|
(5) |
In the next theorem, we give a bijective proof of the conjecture of Stanley by constructing an explicit bijection.
Theorem 9. For positive integer
$ Last(σ)−1=Last(φ(σ)) $
|
(6) |
for all
Given a permutation
$ σ[1]=ˉ4,σ[2]=ˉ4ˉ1,σ[3]=2ˉ4ˉ1,σ[4]=2ˉ4ˉ13,σ[5]=2ˉ4ˉ135. $
|
Some examples of
Definition 10. A type
For example, all type
Our second aim is to show that these two refinements are new Arnold families. Recall that the sequence
$ {{\mathcal{S}}_{n,k}} : = \left\{{\sigma\in {{\mathcal{DU} }_n}^{(B)} : \sigma_1 = k }\right\}. $ |
Theorem 11. For all
$ ψB:Sn,k→T(B)n,k, $
|
(7) |
$ ωB:T(B)n,k→A(B)n,k. $
|
(8) |
Thus, for all
$ Sn,k=#A(B)n,k=#T(B)n,k. $
|
(9) |
In particular, the two sequences
Hetyei[12,Definition 8] defined another class of signed André permutations.
Definition 12 (Hetyei). A signed André permutation is a pair
We write
Conjecture 13. For all
$ Sn,k=#A(H)n+1,n+2−k. $
|
Since the last entry of any permutation in the family
Definition 14. A permutation
Let
Theorem 15. For positive integer
$ Last(σ)−1=Last(φ(B)(σ)) $
|
(10) |
for all
Remark. Ehrenborg and Readdy [6,Section 7] gave a different definition of signed Simsun permutation as follows: A signed permutation
$ 12,\; 21, \; \bar{1}2, \; 2\bar{1}, \; 1\bar{2}, \; \bar{2}1, \; \bar{1}\bar{2}, \; \bar{2}\bar{1} $ |
are Simsun permutations, we note that it is not an Arnold family.
First of all, we prove Theorem 7, in order to show that
Given
For example, if the tree
$ 12,\; 21, \; \bar{1}2, \; 2\bar{1}, \; 1\bar{2}, \; \bar{2}1, \; \bar{1}\bar{2}, \; \bar{2}\bar{1} $ |
then
Given
$ πi={σi−1ifi∉{i1,…,iℓ},σik−1ifi=ik−1fork=2,…,ℓ. $
|
(11) |
We show that
$ \sigma_a = \pi_a+1,\; \sigma_{a+1} = \pi_{a+1}+1,\; \ldots, \sigma_{c-1} = \pi_{c-1}+1 \text{, and } \sigma_{c} \le \pi_{c}+1. $ |
Hence a triple
Consider the running example
Remark. Considering the bijection
$ ψ(τ)=T∈T9,7,ω(T)=σ=684512937∈A9,7,φ(σ)=π=57341286∈RS8,6, $
|
where
One can extend the above mapping
Remark. This bijection preserves the
$ \texttt{ababaaba} = \texttt{cddcd}. $ |
For the cd-index of a Simsun permutation
$ \texttt{aababaab} = \texttt{cddcd}. $ |
Given a
$ \psi^{B}(\sigma) = \pi^{-1}(\psi(\pi \sigma)) $ |
through the unique order-preserving map
For example, in the case of
$ \pi = {\bar{8}\bar{4}\bar{3}\bar{1}25679 \choose 123456789}. $ |
So we have
$ \pi = {\bar{8}\bar{4}\bar{3}\bar{1}25679 \choose 123456789}. $ |
In Subsection 3.1, we define the bijection
$ \omega^{B}(T) = \pi^{-1}(\omega(\pi(T))) $ |
through the unique order-preserving map
For example, in the case of
$ \omega^{B}(T) = \pi^{-1}(\omega(\pi(T))) $ |
we obtain
We summarize four interpretations for Entringer numbers
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In 2012, Chuang et al. [4] construct a bijection
Algorithm A.
(A1) If
(A2) Otherwise, the word
$ \rho(T) = \omega\cdot \rho(T'), $ |
where the subword
(a) If the root of
(b) If the root of
As deleted only
Remark. Originally, in [4], the increasing 1-2 trees on
Theorem 16. The bijection
Proof. Suppose that we let
The root of
To record the left-child
It is clear that all vertices in the minimal path in a tree become the right-to-left minimums in a permutation under
The bijection
Given an increasing 1-2 tree
Algorithm B. Gelineau et al. described the bijection
$ \left|{{{\mathcal{DU} }}_n}\right| = \left|{{\mathcal{T}}_n}\right| = 1, $ |
we can define trivially
(B1) If
$ \pi'_{j} = {πj+2,if πj+2<k−1,πj+2−2,if πj+2>k. $
|
We get
$ \pi'_{j} = {πj+2,if πj+2<k−1,πj+2−2,if πj+2>k. $
|
We get the tree
(B2) If
(a) If
$ \pi'_{j} = {πj+2,if πj+2<k−1,πj+2−2,if πj+2>k. $
|
(b) If
$ \pi'_{j} = {πj+2,if πj+2<k−1,πj+2−2,if πj+2>k. $
|
Algorithm C. We define another bijection
If
For
(C1)
$ v1<u1<v2<u2<⋯<vj−1<uj−1<vj $
|
Decomposing by the maximal path from
● Graft
● Flip the tree at vertex
● Transplant the trees
● Graft
We can illustrate the above transformation by
$ v1<u1<v2<u2<⋯<vj−1<uj−1<vj $
|
(C2) If
● Graft
● Transplant the trees
● Graft
We can illustrate this transformation by the following
$ v1<u1<v2<u2<⋯<vj−1<uj−1<vj $
|
We note that the vertex
Example. We run the new algorithm to the examples
$ d5(σ)=(3),d4(σ)=(6,2),d3(σ)=(9,1),d2(σ)=(8,5),d1(σ)=(7,4). $
|
By Algorithm C, we get five trees sequentially
$ d5(σ)=(3),d4(σ)=(6,2),d3(σ)=(9,1),d2(σ)=(8,5),d1(σ)=(7,4). $
|
with
$ a(4)=3,a(3)=2,a(2)=9,a(1)=5,b(4)=3,b(3)=2,b(2) does not exist,b(1)=5. $
|
Thus, the increasing 1-2 tree
Theorem 17. The two bijections
Proof. It is clear that (C2) is equivalent to (B1). Since the rule (B2a) just exchange two labels, but does not change the tree-structure, it is enough to show that (C1) is produced recursively from (B1) and (B2b).
Assume that
$ a(4)=3,a(3)=2,a(2)=9,a(1)=5,b(4)=3,b(3)=2,b(2) does not exist,b(1)=5. $
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Due to
$ a(4)=3,a(3)=2,a(2)=9,a(1)=5,b(4)=3,b(3)=2,b(2) does not exist,b(1)=5. $
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Since
$ a(4)=3,a(3)=2,a(2)=9,a(1)=5,b(4)=3,b(3)=2,b(2) does not exist,b(1)=5. $
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Since (C2a) is produced from the rule (B1) and (B2b), then Algorithm C follows Algorithm B.
The first author's work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2017R1C1B2008269).
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1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
1 | 1 | 1 | ||||||
2 | 0 | 1 | 1 | |||||
3 | 0 | 1 | 1 | 2 | ||||
4 | 0 | 1 | 2 | 2 | 5 | |||
5 | 0 | 2 | 4 | 5 | 5 | 16 | ||
6 | 0 | 5 | 10 | 14 | 16 | 16 | 61 | |
7 | 0 | 16 | 32 | 46 | 56 | 61 | 61 | 271 |
-6 | -5 | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 | 5 | 6 | ||
1 | 1 | 1 | |||||||||||
2 | 0 | 1 | 1 | 2 | |||||||||
3 | 0 | 2 | 3 | 3 | 4 | 4 | |||||||
4 | 0 | 4 | 8 | 11 | 11 | 14 | 16 | 16 | |||||
5 | 0 | 16 | 32 | 46 | 57 | 57 | 68 | 76 | 80 | 80 | |||
6 | 0 | 80 | 160 | 236 | 304 | 361 | 361 | 418 | 464 | 496 | 512 | 512 |
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1 | 2 | 3 | 4 | 5 | 6 | 7 | ||
1 | 1 | 1 | ||||||
2 | 0 | 1 | 1 | |||||
3 | 0 | 1 | 1 | 2 | ||||
4 | 0 | 1 | 2 | 2 | 5 | |||
5 | 0 | 2 | 4 | 5 | 5 | 16 | ||
6 | 0 | 5 | 10 | 14 | 16 | 16 | 61 | |
7 | 0 | 16 | 32 | 46 | 56 | 61 | 61 | 271 |
-6 | -5 | -4 | -3 | -2 | -1 | 1 | 2 | 3 | 4 | 5 | 6 | ||
1 | 1 | 1 | |||||||||||
2 | 0 | 1 | 1 | 2 | |||||||||
3 | 0 | 2 | 3 | 3 | 4 | 4 | |||||||
4 | 0 | 4 | 8 | 11 | 11 | 14 | 16 | 16 | |||||
5 | 0 | 16 | 32 | 46 | 57 | 57 | 68 | 76 | 80 | 80 | |||
6 | 0 | 80 | 160 | 236 | 304 | 361 | 361 | 418 | 464 | 496 | 512 | 512 |
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