1. Enumerative Combinatorics
Some Topics in
Enumerative Combinatorics
演讲者: 叶永南教授
中研院数学研究所研究员
April 16
2016长三角会议

2. M. P. Schutzenberger, (‘ertain elementary families of automata, in: Proc. Symp. on Mathematical Theory of Automata
(Polytechnic Institute of Brooklyn, 1962) pp. I
M.P. Schutzenberger, Context-free languages and pushdown automata, Inormation and Control 6 ( I963 )

3. Classical examples are those used with enumeration of trees or related objects
and can be found in
Deep-going examples are found in the work of Cori and Vauquelin

4. Motzkin paths
An n-Motizkin path is a lattice path from (0,0) to (n,0) in the plane integer lattice Z x Z consisting of up-step (1,1), level-step (1,0) and down-step (1,-1), which never passes below the x-axis.
(0,0)
(17,0)

7. Classically, from the non-ambiguous grammar, one can associate a proper algebraic system of equations in noncommutative power series. The unique solution of system contains the (noncommutative) generating function of the language L. By sending all variables x of X onto one variable t, the series L becomes solution of an algebraic system in one variable t .
Usually an explicit formula is known for
by means of classical calculus techniques used in combinatorics (recurrence relation, Lagrange inversion formula, etc.).

9. A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling with a connected interior

11. Klarner gives the expression for the generating function enumerating row-convex polyominoes according to area.
D. KLARNER, Some results concerning polyominoes, Fibonacci Quart. 3 (1965), 9-20.
Delest and Viennot found an exact formula for the number p2,, of convex polyominoes with perimeter 2n. The method used, due to ideas of Schtitzenberger, is in three steps:

15. [l] E. Barcucci, A. Del Lungo, R. Pinzani and R
[l] E. Barcucci, A. Del Lungo, R. Pinzani and R. Sprugnoli, La hauteur des polyominos diriges verticalement convexes, in: Actes du 31eme Seminaire Lotharingien de Combinatoire, Publi. IRMA, Universite Strasbourg I, 1993.
[2] E. Barcucci, R. Pinzani and R. Sprugnoli, Generation al&atone des animaux dirigts, in: J. Labelle et J.G. Penaud, eds., Publications du LACIM 10, Universite du Quebec a Montreal (1991)
[3] E. Barcucci, R. Pinzani and R. Sprugnoli, Directed column-convex polyominoes by recurrence relations, in: M.C. Gaudel, J.P. Jouamtaud, eds., TAPSOFT’93. Lecture Notes in computer Science, Vol. 668 (Springer, Berlin, 1993)
[4] E. Barcucci, R. Pinzani and R. Sprugnoli, The random generation of directed animals, Theoret. Comput. Sci. 127 (1994)
[5] G.D. Birkhoff, Formal theory of irregular difference equations, Acta Math. 54 (1939)
[6] M. Bousquet-MClou, q&numeration de polyominos convexes, Publications du LACIM 9, Universiti du Quebec a Montreal, 1991.
[7] M. Delest, Polyominoes and animals: some recent results, J. Comput. Chem. 8 (1991) 3-18.
[8] M. Delest and S. Dulucq, Enumeration of directed column-convex animals with given perimeter and area, Rapport LaBRI 86-15, Universitt Bordeaux 1, 1987.
[9] M. Delest and X.G. Viennot, Algebraic languages and polyominoes enumeration, Theoret. Comput. Sci., 34 (1984)
[lo] B. Derrida, J.P. Nadal and J. Vannimenus, Directed lattice animals in 2 dimensions: numerical and exact results, J. Physique 43 (1982) 1561.

16. [ll] D. Dhar, M. Phani and M. Barma, Enumeration of directed site animals on two-dimensional lattices, J. Phys. A 15 (1982) L279-L284.
[12] D. Gouyou-Beauchamps and X.G. Viennot, Equivalence of the two-dimensional directed animal problem to a one-dimensional path problem, Adu. Appl. Math. 9 (1988)
[13] L.R. Graham, D.E. Knuth and 0. Patashnik, Concrete Mathematics (Addison-Wesley, Reading, MA, 1989).
[14] J.G. Penaud, Une nouvelle bijection pour les animaux dirigts, Acres du 22eme Seminaire Lotharingien de Combinatoire (Publi. IRMA, Universite Strasbourg I, 1989).
[15] M.P. Schiltzenberger, Context-free languages and pushdown automata, Inform. and Control 6 (1963)
[16] N.J.A. Sloane, A Handbook of Integer Sequences (Academic Press, New York, 1973).
[17] X.G. Viennot, Problbmes combinatoires poses par la physique statistique, Seminaire Bourbaki no 626, 36eme an&e, Asterisque, Vol (1985)
[18] X.G. Viennot, A Survey of polyomino enumeration, in: P. Leroux et C. Reutenauer, eds., Proc. Series jormelles et combinatoire algebrique, Publications du LACIM 11, Universit& du Quebec B Montreal,
1992.

17. A. JOYAL, Une theorie combinatoire des series formelles, Adv. in Math
A. JOYAL, Une theorie combinatoire des series formelles, Adv. in Math. 42 (1981), l-82.

22. [1] F. Bergeron, Une combinatoire du plethysme, Journal of the Combinatorial Theory, Series A 46 (1987) 291–305. ´
[4] F. Bergeron, G. Labelle, P. Leroux, Combinatorial Species and Tree-Like Structures, in: Encyclopedia of Mathematics and its Applications,
vol. 67, Cambridge University Press, Cambridge, 1998.
[5] [15] A. Joyal, Une theorie combinatoire des s ´ eries formelles, Advances in Mathematics 42 (1981) 1–82. ´
[16] G. Labelle, On asymmetric structures, Discrete Mathematics 99 (1992) 141–164.
[17] G. Labelle, P. Leroux, An extension of the exponential formula in enumerative combinatorics, Electronic Journal of Combinatorics 3 (2)
(1996) #R12.
[18] N.C. Metropolis, G.C. Rota, Witt vectors and the algebra of necklaces, Advances in Mathematics 50 (1983) 95–125.
[19] O. Nava, G.C. Rota, Plethysm, categories and combinatorics, Advances in Mathematics 58 (1985) 61–68.
[20] B. Pittel, Where the typical set partitions meet and join, Electronic Journal of Combinatorics 7 (2000) #R5.
[21] R.P. Stanley, Enumerative Combinatorics, vol. 2, Wadsworth Brooks/Cole, Pacific Grove, CA, 1986.
[22] D.E. Taylor, A natural proof of cyclotomic identity, Bulletin of the Australian Mathematical Society 42 (1990) 185–189.
[23] H.S. Wilf, Generatingfunctionology, Academic Press, Boston, 1994.
[24] Y.N. Yeh, On the combinatorial species of Joyal, Ph.D. Dissertation, State University of New York at Buffalo, 1985.
[25] Y.N. Yeh, The calculus of virtual species and K-species, in: G. Labelle, P. Leroux (Eds.), Combinatoire Enum ´ erative, in: Lecture Notes in ´
Mathematics, vol. 1234, Springer-Verlag, Berlin, Heidelberg, New York, 1986, pp. 351–369
W.Y.C. Chen, The Theory of Compositionals, Discrete Mathematics, 22, 1993, 59–87.
W.Y.C. Chen, Compositional Calculus, Journal of Combinatorial Theory, Series A, 64,1993, 149–188.
W.Y.C. Chen, A Bijection for Enriched Trees, European Journal of Combinatorics, 15,
1994, 337–343.
G. Labelle, Sur l’inversion et l’it´eration continue des s´eries formelles, European Journal of Combinatorics, 1, 1980, 113–138.
G. Labelle, Une nouvelle d´emonstration combinatoire des formules d’inversion de Lagrange, Advances in Mathematics, 42, 1981, 217–247.
G. Labelle, Une combinatoire sous-jacente au th´eor`eme des fonctions implicites, Journal of Combinatorial Theory, Series A, 40, 1985, 377–393.
G. Labelle, Eclosions combinatoires appliqu´ees `a l’inversion multidimensionnelle des s´eries formelles, Journal of Combinatorial Theory, Series A, 39, 1985, 52–82.
G. Labelle, Some New Computational Methods in the Theory of Species, in [196], 1986,
192–209.
G. Labelle, Interpolation dans les K-esp`eces, in Actes du s´eminaire lotharingien de combinatoire,14e session, Ed. V. Strehl, Publications de l’Institut de recherche math´ematiques, Strasbourg, France, 1986, 60–70.
G. Labelle, On Combinatorial Differential Equations, Journal of Mathematical Analysis
Applications, 113, 1986, 344–381.
G. Labelle, D´eriv´ees directionnelles et d´eveloppements de Taylor combinatoires, Discrete Mathematics, 79, 1989, 279–297.
G. Labelle, On the Generalized Iterates of Yeh’s Combinatorial K-Species, Journal of
Combinatorial Theory, Series A, 50, 1989, 235–258.

23. (with F. Bergeron) "The factoriality of the ring of S-species", Journal of combinatorial Theory (series A), 55( ),      
(with J. Labelle) "Generalized Dyck paths", Discrete Mathematics, 82 ( ), 1-6.       
(with J. Labelle) "Combinatorial proof of some limit formulas involving Orthogonal polynomials", Discrete Mathematics, 79( ),      
(with J. Labelle) "Dyck paths of knight moves", Discrete Applied Mathematics, 24( ),        
(with J. Labelle) "The relations between permutation groups and combinatorics species", Journal of Combinatorial Theory (series A), 50( ),        
(with J. Labelle) "The combinatorics of Laguerre, Charlier and Hermite polynomials revisited", Studies in Applied Mathematics, 80( ), no. 1,        
(with J. Labelle) "Some combinatorics of the hypergeometric series", European Journal of Combinatorics, 6( ),        
"The calculus of virtual species an K-species", Lecture Notes in Mathematics, Springer Verlag, 1234(1986),        

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30. Hermite polynomials were defined by Laplace (1810) [1] though in scarcely recognizable form, and studied in detail by Chebyshev (1859).[2] Chebyshev's work was overlooked and they were named later after Charles Hermite who wrote on the polynomials in 1864 describing them as new.[3] They were consequently not new although in later 1865 papers Hermite was the first to define the multidimensional polynomials.

32. 32

33. In mathematics, the Laguerre polynomials , named after Edmond Laguerre ( ), are solutions of Laguerre's equation:
which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.
generalized Laguerre polynomials

34.  The Rodrigues formula,

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78. (1-u+tu ^2)G(u) = u(1-u)+tu^2G(1)
1-u+tu ^2 = 0  0 = u(1-u)+tu^2G(1)
 G(1) = u

79. 第79页

82. (2,3)

95. 1: Stirling permutations, cycle structure of permutations and perfect matchings (Electronic journal of combinatorics 22(4) (2015), #P4.42)
Abstract: In this paper, we study the relationship between Stirling permutations, the cycle structure of permutations and perfect matchings, and will give constructive proofs for the equidistribution of some combinatorial statistics on these combinatorial structures.

96. (1) Basic definitions For example, ap(221133)=2. References:
[6] I. Gessel and R.P. Stanley. Stirling polynomials. J. Combin. Theory Ser. A, 24:25-33, 1978.
[11] S.-M. Ma, T. Mansour. The 1/k-Eulerian polynomials and k-Stirling permutations. Discrete Math., 338: , 2015.

97. The excedance statistic on symmetric group

98. The even maximal elements of perfect matchings
A perfect matching M of [2n] is a set partition of [2n] with blocks (disjoint nonempty subsets) of size exactly 2.
We say (a,b) a block of M with even maximal element if the larger element of this block is even.
Let mark(M) be the number of even maximal elements of M.
For example, mark((1,3)(2,4)(5,6))=2.
Let be the set of matchings of [2n].

99. The SPM (Stirling-Permutations-Matchings) sequences

101. (2) Main result
Theorem 1.

102. (3) Constructive proofs of Theorem 1
A bijection between SPM-sequences and perfect matchings

103. (3) Constructive proofs of Theorem 1
A bijection between SPM-sequences and Stirling permutations

104. (3) Constructive proofs of Theorem 1
A map from SPM-sequences to permutations

105. 2. The peak statistics on simsun permutations (Electronic journal of combinatorics 23(2) (2016), #P2.14)
Abstract: In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations.
Properties of the generating polynomials, including the recurrence relation, generating function and real-rootedness are studied. Moreover, we introduce and study simsun permutations of the second kind.

106. (1) Basic definitions
For example , is a simsun permutation, but is not.

107. Simsun permutations and Euler numbers
Let be set of simsun permutations of length n.
Simion and Sundaram discovered that
Reference:
S. Sundaram. The homology representations of the symmetric group on Cohen-
Macaulay subposets of the partition lattice. Adv. Math., 104(2): , 1994.

108. The descent statistic on simsun permutations
Let
Chow and Shiu obtained that
Reference
C-O. Chow, W. C. Shiu. Counting simsun permutations by descents. Ann. Comb.,
15: , 2011.

109. Relationship to left peaks of permutations
A left peak in is an index such that
Let lpk( ) denote the number of left peaks in .
Note that any descent of a simsun permutation is a left peak of itself. We have

110. (2) Relationship to left peaks of permutations
Let
It is well known that
We observed that
Equivalently,

111. (3) Relationship to interior peaks of permutations
Recall that
It is natural to establish a connection between W(n,k) and S(n,k).
For example, W(5,0)=16, W(5,1)=88,W(5,2)=16.
S(4,0)= 1, S(4,1)=11,S(4,2)=4.

112. (3) Relationship to interior peaks of permutations
We present a constructive proof of

113. (3) Relationship to interior peaks of permutations
A constructive proof of
We construct a correspondence,

114. (4) Interior peaks of simsun permutations
Theorem 2.

115. (5) Polynomials with only real zeros

116. (5) Polynomials with only real zeros
Theorem 3.

117. (6) Up-down runs of simsun permutations

118. (6) Up-down runs of simsun permutations
Theorem 4.
where

119. (7) Simsun permutations of the second kind

120. (7) Simsun permutations of the second kind
We present a constructive proof of the following identity :

121. (7) Simsun permutations of the second kind
Let
Theorem 5. We have
where
and

122. 3: Simsun successions and simsun patterns
Abstract: In this paper, we introduce the defin- itions of simsun succession, simsun cycle succession and simsun patterns. In particular, the ordinary simsun permutations are permutations avoiding simsun pattern 321. We present a bijection between permutations avoiding simsun pattern 132 and set partitions.

123. (1) A subset of the set of simsun permutations with no successions
Let

124. (2) Simsun cycle successions

125. (2) Simsun cycle successions
Theorem 5.

126. (3) Permutations avoiding the simsun pattern 132 and set partitions (basic definitions)

127. (3) Permutations avoiding the simsun pattern 132 and set partitions (basic definitions)

128. (3) Permutations avoiding the simsun pattern 132 and set partitions (basic definitions)

129. (3) Permutations avoiding the simsun pattern 132 and set partitions (Main results)
Theorem 6.

130. (3) Permutations avoiding the simsun pattern 132 and set partitions (Proof of Theorem 6)

131. (3) Permutations avoiding the simsun pattern 132 and set partitions (Proof of Theorem 6)
We construct a bijection,

132. (3) Permutations avoiding the simsun pattern 132 and set partitions (Proof of Theorem 6)

133. 4. Enumeration of permutations by number of alternating descents (Discrete Mathematics 339 (2016) 1362–1367)
Abstract. In this paper we present an explicit formula for the number of permutations with a given number of alternating descents. As an application, we obtain the interlacing property of the real and imaginary parts of the zeros of the generating polynomials of these numbers.

134. (1) Basic definitions References:
[2] D. Chebikin, Variations on descents and inversions in permutations, Electron. J. Combin. 15 (2008) #R132.

135. (1) Basic definitions

136. (2) Derivative polynomials of tangent function

137. (3) An Identity
Theorem 1.
As an application, we get

138. (4) The interlacing property of zeros of alternating Eulerian polynomials

139. (5) Alternating descents of signed permutations

140. (6) Derivative polynomials of secant functions

141. 5. The cycle descent statistic on permutations
Abstract: In this paper we study the cycle descent statistic on permutations. Several involutions on permutations and derangements are constructed. Moreover, we present a bijection between permutations and perfect matchings.

143. (1) Basic definitions of cycle descents

144. (1) Basic definitions of negative cycle-descent permutations

145. (1) Basic definitions

146. (2) Recurrence relations
the

147. (3) Perfect matchings

148. (3) Perfect matchings

149. Chen, Xiao-Min; Chang, Xiang-Ke; Sun, Jian-Qing; Hu, Xing-Biao; Yeh, Yeong-nan
"Three semi-discrete integrable systems related to orthogonal polynomials and their generalized determinant solutions.", Nonlinearity, 28(2015),

150. In this paper, we particularly focus on three semi-discrete integrable systems, which have direct links with orthogonal polynomials.
The first one is the Schur flow. Mukaihira and Nakamura discussed monic orthogonal polynomials on the unit circle, which are named as the Szeg˝o orthogonal polynomials.
The second one is related to the relativistic Toda chain, which has intimate connections with the Laurent biorthogonal polynomials
The last semi-discrete integrable system discussed in this paper is the nonisospectral Toda lattice, which is in the framework of orthogonal polynomials as well.
Aiming at the above three semi-discrete integrable systems, we design to seek their generalized determinant solutions respectively。Kajiwara et al introduced the convolution term for the evolution of moments in the Hankel determinants:
with two arbitrary functions φ(t) and ψ(t).

157. Xiang-Ke Chang, Xing-Biao Hu, Hongchuan Lei, Yeong-Nan Yeh
"Combinatorial proofs of addition formulas", Electronic Journal of Combinatorics, (2016).

165. Thank you
for your attention!