高雄大學游森棚教授的網頁, 高等數學分頁
Sen-Peng Eu
Associate professor, National University of Kaohsiung, Taiwan

(A) 學術期刊論文(Journal papers) (Updated Apr/10/2007)

1. Sen-Peng Eu, and Tung-Shan Fu (2007), Lattice Paths and Generalized Cluster Complexes, accepted by Journal of Combinatorial Theory, Series A (NSC-95-2115-M-390-005-MY3) [SCI, NUK]

In this paper we propose a variant of the generalized Schr\"oder paths and generalized Delannoy paths by giving a restriction on the positions of certain steps. This generalization turns out to be reasonable, as attested by the connection with the faces of generalized cluster complexes of type $A$ and $B$. As a result, we derive Krattenthaler's $F$-triangles for these two types by a combinatorial approach in terms of lattice paths.

2. Sen-Peng Eu, and Tung-Shan Fu (2007), The Cyclic Sieving Phenomenon for Faces of Generalized Cluster Complexes, accepted by Advances in Applied Mathematics (NSC-95-2115-M-390-005-MY3) [SCI, NUK]

The notion of cyclic sieving phenomenon was introduced by Reiner, Stanton, and White as a generalization of Stembridge's $q=-1$ phenomenon. The generalized cluster complexes associated to root systems are given by Fomin and Reading as a generalization of the cluster complexes found by Fomin and Zelevinsky. In this paper, the faces of various dimensions of the generalized cluster complexes in type $A_n$, $B_n$, $D_n$, and $I_2(a)$ are shown to exhibit the cyclic sieving phenomenon under a cyclic group action. For the cluster complexes of exceptional type $E_6$, $E_7$, $E_8$, $F_4$, $H_3$, and $H_4$, a verification for such a phenomenon on the facets is given.

3. Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh (2007), Catalan and Motzkin numbers mod 4 and 8, accepted by European Journal of Combinatorics. (NSC-95-2115-M-390-005-MY3) [SCI, NUK]

In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8.

4. Gerald J. Chang, Sen-Peng Eu, and Chung-Heng Yeh (2007), On the antipodal Gray codes, to appear in Theoretical Computer Science (374), 2007, 82-90 (NSC-95-2115-M-390-005-MY3) [SCI, NUK]

An $n$-bit Gray code is a circular listing of all $2^n$ $n$-bit binary strings in which consecutive strings differ at exactly one bit. For $n \le t \le 2^{n-1}$, an $(n,t)$-antipodal Gray code is a Gray code in which the complement of any string appears $t$ steps away from the string, clockwise or counterclockwise. Killian and Savage proved that an $(n,n)$-antipodal Gray code exists when $n$ is a power of $2$ or $n=3$, and does not exist for $n=6$ or odd $n > 3$. Motivated by these results, we prove that for odd $n \ge 3$, an $(n,t)$-antipodal Gray code exists if and only if $t=2^{n-1}-1$. For even $n$, we establish two recursive constructions for $(n,t)$ codes from smaller $(n',t')$. Consequently, various $(n,t)$-antipodal Gray codes are found for even $n$'s. Examples are for $t=2^{n-1}-2^k$ with $k$ odd and $1 \le k \le n-3$ when $n \ge 4$, for $t=2^{n-k}$ when $n \ge 2k$ with $1 \le k \le 3$, for $t = n$ when $n = 2^k \ge 2$ (an alternative proof for Killian and Savage's result) $\ldots$ etc.

5. Szu-En Cheng, Sen-Peng Eu, and Tung-Shan Fu (2006), Area of Catalan Paths on Checkerboard, accepted by European Journal of Combinatorics (NSC-94-2115-M-390-005) [SCI, NUK]

It is known that the area of all Catalan paths of length n is equal to the number of inversions of all 321-avoiding permutations of length n+1. In this paper, a bijection between the two sets is established. Meanwhile, a number of interesting bijective results that pave the way to the required bijection are presented.

6. Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh (2006), On the Congruences of Some Combinatorial Numbers, Studies in Applied Mathematics (116), 2006, 135-144. (NSC-93-2115-M-390-005) [SCI, NUK]

In this paper, we apply Lucas’ theorem to evaluate the congruences of several combinatorial numbers, including the central Delannoy numbers and a class of Ap悶ry-like numbers, the numbers of noncrossing connected graphs, the numbers of total edges of all noncrossing connected graphs on n vertices, etc. One of these results verifies a conjecture given by Deutsch and Sagan recently. In the end, we use an automaton to explain the idea of our approach.

7. Sen-Peng Eu, Bo-In Yang and Yeong-Nan Yeh (2006), Generalized Wiener Indices in Hexagonal Chains, International Journal of Quantum Chemistry.(106-2), 426-435 (NSC-93-2115-M-390-005) [SCI, NUK]

The Wiener index, or the Wiener number, also known as the “sum of distances” of a connected graph, is one of the quantities associated with a molecular graph that correlates nicely to physical and chemical properties, and has been studied in depth. An index proposed by Schultz is shown to be related to the Wiener index for trees, and Ivan Gutman proposed a modification of the Schultz index with similar properties. We deduce a similar relationship between these three indices for catacondensed benzenoid hydrocarbons (graphs formed of concatenated hexagons, or hexagonal chains, or sometimes acenes). Indeed, we may define three families of generalized Wiener indices, which include the Schultz and Modified Schultz indices as special cases, such that similar explicit formulae for all generalized Wiener indices hold on hexagonal chains. We accomplish this by first giving a more refined proof of the formula for the standard Wiener index of a hexagonal chain, then extending it to the generalized Wiener indices via the notion of partial Wiener indices. Finally, we discuss possible extensions of the result.

8. Sen-Peng Eu, Tung-Shan Fu and Chun-Ju Lai (2005), On Enumeration of Parking Functions by Leading Numbers, Advances in Applied Mathematics (35), 2005, 355-442 (NSC-93-2115-M-390-005) [SCI, NUK]

Let x=(x1,x2,...x_n) be a sequence of positive integers. An x-parking function is a sequence (a1,...,an) of positive integers whose non-decreasing rearrangement b1,...bn satisfies b1<=x1+...+xn. In this paper we give a combinatorial approach to the enumeration of (a,b,...,b) parking functions by their leading terms, which covers the special cases x=(1,...,1), (a,1,....,1), and (b,....,b). The approach relies on bijections between the x-parking functions and labeled rooted forests. To serve this purpose, we present a simple method for establishing the required bijections. Some bijective results between certain sets of x-arking functions of distinct leading terms are also given. づ

9. Sen-Peng Eu, Tung-Shan Fu and Yeong-Nan Yeh (2005), Refined Chung-Feller Theorem, accepted by Journal of Combinatorics Theory series A. (NSC-92-2119-M-390-001) [SCI, NUK].

In this paper we prove a strengthening of the classical Chung–Feller theorem and a weighted version for Schroder paths. Both results are proved by refined bijections which are developed from the study of Taylor expansions of generating functions. By the same technique, we establish variants of the bijections for Catalan paths of order d and certain families of Motzkin paths. Moreover, we obtain a neat formula for enumerating Schroder paths づ

10. Sen-Peng Eu and Tung-Shan Fu (2005), A simple proof of the Aztec Diamond Theorem, accepted by Electronic Journal of Combinatorics (12), #R18, 1-8, 2005. (NSC-93-2115-M-390-005) [SCIE, NUK].

Based on a bijection between domino tilings of an Aztec diamond and nonintersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schr?oder numbers. づ

11. Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh (2004), Odd or Even on Plane Trees. Discrete Mathematics (281), 189-196, 2004. (NSC-92-2119-M-390-001) [SCI, NUK].

Over all plane trees with n edges, the total number of vertices with odd degree is twice the number of those with odd outdegree. Deutsch and Shapiro posed the problem of 3nding a direct two-to-one correspondence for this property. In this article, we give three di5erent proofs via generating functions, an inductive proof and a two-to-one correspondence. Besides, we introduce two new sequences which enumerate plane trees according to the parity of the number of leaves. The explicit formulae for these sequences are given. As an application, the relation provides a simple proof for a problem concerning colored nets in Stanley’s Catalan Addendum. づ

12. Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh (2003), Dyck Path with peaks Avoiding/Restricted to an Arbitrary Set , Studies in Applied Mathematics (111), 2003, 453-465 [SCI, NUK] .

To be added soon. づ

13. Sen-Peng Eu, Shu-Chung Liu and Yeong-Nan Yeh, (2002), Taylor Expansion for Catalan and Motzkin Numbers , Advances in Applied Mathematics (29), 2002, 345-357 [SCI, NTNU].

In this paper we introduce two new expansions for the generating functions of Catalan numbers and Motzkin numbers. The novelty of the expansions comes from writing the Taylor remainder as a functional of the generating function. We give combinatorial interpretations of the coefficients of these two expansions and derive several new results. These findings can be used to prove some old formulae associated with Catalan and Motzkin numbers. In particular, our expansion for Catalan number provides a simple proof of the classic Chung–Feller theorem; similar result for the Motzkin paths with flaws is also given. づ

(B) 研討會論文(Conference papers)

(C) 技術報告及其它 (Thesis, Reports & Preprints)

(D) 獲獎及其他(Awards)

(E) 學術計畫 (Research Grants)

(F) 學術活動及演講(Academic Activities and Talks)

(G) 教育活動及演講(Educational Activities and Talks)