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Page 4416 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
Bernard Harris (1-W1-S; Madison, W1) 2001g:05010 OSA15 60C05 Ehrenborg, Richard (S-RIT; Stockholm) A bijective answer to a question of Zvonkin. (English summary) Ann. Comb. 4 (2000), no. 2, 195—197.  ...  Summary: “The purpose of this note is to give a bijective proof of the identity E| Il (Xj — X;)?] =01- 1!  ... 

Page 770 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews  
proofs of a conjecture of Simion.  ...  Similar questions are raised and answered for other s and p. The fractal-like appearance of these triangular arrays is also dis- cussed. G. L.  ... 

Identities from weighted Motzkin paths

William Y.C. Chen, Sherry H.F. Yan, Laura L.M. Yang
2008 Advances in Applied Mathematics  
Based on a weighted version of the bijection between Dyck paths and 2-Motzkin paths, we find combinatorial interpretations of two identities related to the Narayana polynomials and the Catalan numbers.  ...  These interpretations answer two questions posed recently by Coker.  ...  We are also grateful to Joseph P.S. Kung for a careful reading of the manuscript and for helpful comments.  ... 
doi:10.1016/j.aam.2004.11.007 fatcat:kol3w6o2bfa27n5wyj2p46wf54

Stacking Blocks and Counting Permutations

Lara K. Pudwell
2010 Mathematics Magazine  
In this paper we will explore two seemingly unrelated counting questions, both of which are answered by the same formula.  ...  Their web search produced a conjecture that gives a nice geometric interpretation of a permutation patterns question.  ...  Acknowledgment Thank you to David Harris for inspiring this paper, and to Andrew Baxter and two anonymous referees for many valuable presentation suggestions.  ... 
doi:10.4169/002557010x521868 fatcat:rbc4wk2q5vgodkjksuuewxqwga

Pattern avoidance in compositions and multiset permutations [article]

Carla D. Savage, Herbert S. Wilf
2005 arXiv   pre-print
bijective proof of this fact first for the pattern (123), and then for all patterns in S_3 by using a recently discovered bijection of Amy N.  ...  In the former case we determine the generating function explicitly, for integer compositions of n that avoid a given pattern of length 3 and we show that the answer is the same for all such patterns.  ...  However the map Θ = Θ(a, a ′ ) can be composed with the bijections of Myers [4] , which generalize earlier constructions of Simion and Schmidt [6] to give bijective proofs of symmetry for all six patterns  ... 
arXiv:math/0504310v1 fatcat:2pdm6brdgraxtphh24qyfcqwuq

Simion's Type B Associahedron is a Pulling Triangulation of the Legendre Polytope

Richard Ehrenborg, Gábor Hetyei, Margaret Readdy
2018 Discrete & Computational Geometry  
Finally, we present a bijection between the faces of the Simion type B associahedron and Delannoy paths.  ...  We show that the Simion type B associahedron is combinatorially equivalent to a pulling triangulation of the type A root polytope known as the Legendre polytope.  ...  This work was partially supported by grants from the Simons Foundation (#429370 to Richard Ehrenborg, #245153 and #514648 to Gábor Hetyei, #206001 and #422467 to Margaret Readdy).  ... 
doi:10.1007/s00454-018-9973-4 fatcat:jsga4wxegbfc7prkafh74xrvsy

Page 3720 of Mathematical Reviews Vol. , Issue 99f [page]

1999 Mathematical Reviews  
We show that having such an exten- sion is a £|-complete property and so there is no Borel answer to Los’s question.  ...  One may ask whether it is possible to find a rep- resentation (g,q) such that g: A — Z, where Z means the set of all integers. In Chapter 4 the authors give an answer to this problem.  ... 

Page 6422 of Mathematical Reviews Vol. , Issue 95k [page]

1995 Mathematical Reviews  
It is an open question to find an expression for a, valid for all n.  ...  These bijections restrict to non-crossing partitions and so also prove some equidistribution results of R. E. Simion [J. Combin. Theory Ser. A 66 (1994), no. 2, 270-301; MR 95e:05009].  ... 

Direct products and the contravariant hom-functor

Simion Breaz
2011 Bulletin of the London Mathematical Society  
We prove in ZFC that if G is a (right) R-module such that the groups _R(∏_i∈ IG_i,G) and ∏_i∈ I_R(G_i,G) are naturally isomorphic for all families of R-modules (G_i)_i∈ I then G=0.  ...  The result is valid even we restrict to families such that G_i G for all i∈ I.  ...  I would like to thank to Ciprian Modoi and Phill Schultz for illuminating discussions on subjects related to the main result of this note.  ... 
doi:10.1112/blms/bdr083 fatcat:b5ng74mqlzby3o3wd6zrnr2f6e

A Second Look at the Toric h-Polynomial of a Cubical Complex

Gábor Hetyei
2012 Annals of Combinatorics  
By discovering another variant of the Gessel-Shapiro result in the work of Denise and Simion, we find evidence that the toric h-polynomials of cubes are related to the Morgan-Voyce polynomials via Viennot's  ...  We provide an explicit formula for the toric h-contribution of each cubical shelling component, and a new combinatorial model to prove Clara Chan's result on the non-negativity of these contributions.  ...  Thus it may be worthwhile to generalize the Denise-Simion colored Motzkin path enumeration problem for f d (0, 0, x) to questions whose answers are given by the polynomials f d (i, j, x).  ... 
doi:10.1007/s00026-012-0144-7 fatcat:4na4pdzcxvgddj67cmiohzmp4y

Simion's type B associahedron is a pulling triangulation of the Legendre polytope [article]

Richard Ehrenborg, Gabor Hetyei, Margaret Readdy
2016 arXiv   pre-print
Finally, we present a bijection between the faces of the Simion's type B associahedron and Delannoy paths.  ...  We extend Cho's cyclic group action to the triangulation in such a way that it corresponds to rotating centrally symmetric triangulations of a regular (2n+2)-gon.  ...  This work was partially supported by two grants from the Simons Foundation (#245153 to Gábor Hetyei and #206001 to Margaret Readdy).  ... 
arXiv:1607.06061v1 fatcat:b4tcjqjoxbfclg3r2mxxjwvmym

A new matching property for posets and existence of disjoint chains

Mark J. Logan, Shahriar Shahriari
2004 Journal of combinatorial theory. Series A  
lattice, the lattice of partitions of a finite set, the intersection poset of a central hyperplane arrangement, the face lattice of a convex polytope, the lattice of noncrossing partitions, and any geometric  ...  One complication is that it may not be possible to have the chains respect the original matching and hence, in the constructed set of chains, x i and y i may not be in the same chain.  ...  elements of B to elements of C; and from elements of C to elements of A; precisely when the elements in question are comparable in P: We will show that G is q-connected from s to t; that is, removal of  ... 
doi:10.1016/j.jcta.2004.06.002 fatcat:soe7g4jy6zcz7gcdxm5euzrw6i

Complexity problems in enumerative combinatorics [article]

Igor Pak
2018 arXiv   pre-print
We give a broad survey of recent results in Enumerative Combinatorics and their complexity aspects.  ...  Ira Gessel kindly suggested the proof of Theorem 1.4.  ...  Special thanks to Stephen DeSalvo, Scott Garrabrant, Alejandro Morales, Danny Nguyen, Greta Panova, Jed Yang and Damir Yeliussizov for many collaborations and numerous discussions, some of which undoubtedly  ... 
arXiv:1803.06636v2 fatcat:qrnq7kmr45do5oeumxneccovma

The patterns of permutations

Herbert S. Wilf
2002 Discrete Mathematics  
A pattern is said to occur in a permutation if there are integers 1 6 i1 ¡ i2 ¡ · · · ¡ i k 6 n such that for all 1 6 r ¡ s 6 k we have (r) ¡ (s) if and only if (ir) ¡ (is). Example. Suppose = (132).  ...  Let n; k be positive integers, with k 6 n, and let be a ÿxed permutation of {1; : : : ; k}. 1 We will call the pattern. We will look for the pattern in permutations of n letters.  ...  BÃ ona [3] answered a rmatively for = (132) and all r. Richard Stanley is inclined to favor a negative answer to N -Z's question.  ... 
doi:10.1016/s0012-365x(02)00515-0 fatcat:4mfhdis5abeeffwcw6jvxccfgm

Folding Phenomenon of Major-balance Identities on Restricted Involutions [article]

Tung-Shan Fu, Hsian-Chun Hsu, Hsin-Chieh Liao
2017 arXiv   pre-print
Moreover, we prove affirmatively a question about refined major-balance identity on the 123-avoiding involutions, respecting the number of descents.  ...  In this paper we prove a refined major-balance identity on the 321-avoiding involutions of length n, respecting the leading element of permutations.  ...  In addition to answering the above question, one of the main results in this paper is the following enumeration of joint distributions for two statistics of 321-avoiding involutions. Theorem 1.4.  ... 
arXiv:1711.04601v1 fatcat:pk4lo3sljzcrref3wspnnumru4
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