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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! ...##
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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. ...##
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Identities from weighted Motzkin paths

2008
*
Advances in Applied Mathematics
*

Based on

doi:10.1016/j.aam.2004.11.007
fatcat:kol3w6o2bfa27n5wyj2p46wf54
*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. ...##
###
Stacking Blocks and Counting Permutations

2010
*
Mathematics Magazine
*

In this paper we will explore two seemingly unrelated counting

doi:10.4169/002557010x521868
fatcat:rbc4wk2q5vgodkjksuuewxqwga
*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. ...##
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Pattern avoidance in compositions and multiset permutations
[article]

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 ...

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

2018
*
Discrete & Computational Geometry
*

Finally, we present

doi:10.1007/s00454-018-9973-4
fatcat:jsga4wxegbfc7prkafh74xrvsy
*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). ...##
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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. ...##
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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]. ...##
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Direct products and the contravariant hom-functor

2011
*
Bulletin of the London Mathematical Society
*

We prove in ZFC that if G is

doi:10.1112/blms/bdr083
fatcat:b5ng74mqlzby3o3wd6zrnr2f6e
*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. ...##
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A Second Look at the Toric h-Polynomial of a Cubical Complex

2012
*
Annals of Combinatorics
*

By discovering another variant

doi:10.1007/s00026-012-0144-7
fatcat:4na4pdzcxvgddj67cmiohzmp4y
*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). ...##
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Simion's type B associahedron is a pulling triangulation of the Legendre polytope
[article]

2016
*
arXiv
*
pre-print

Finally, we present

arXiv:1607.06061v1
fatcat:b4tcjqjoxbfclg3r2mxxjwvmym
*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). ...##
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A new matching property for posets and existence of disjoint chains

2004
*
Journal of combinatorial theory. Series A
*

lattice, the lattice

doi:10.1016/j.jcta.2004.06.002
fatcat:soe7g4jy6zcz7gcdxm5euzrw6i
*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*...##
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Complexity problems in enumerative combinatorics
[article]

2018
*
arXiv
*
pre-print

We give

arXiv:1803.06636v2
fatcat:qrnq7kmr45do5oeumxneccovma
*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 ...##
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The patterns of permutations

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*. ...

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

2017
*
arXiv
*
pre-print

Moreover, we prove affirmatively

arXiv:1711.04601v1
fatcat:pk4lo3sljzcrref3wspnnumru4
*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. ...
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