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Reduced matrices and q-log-concavity properties of q-Stirling numbers

Pierre Leroux
1990 Journal of combinatorial theory. Series A  
Other related results, methods, counterexamples, and conjectures are discussed. ci;'  ...  Butler's method which is based on involutions on pairs of Ferrers diagrams [ Bu2] .  ...  RESULTS AND CONJECTURES In this section we present a summary of results (with proofs), counterexamples, and conjectures, dealing with log-concavity and unimodality of q-and p, q-analogues of Stirling numbers  ... 
doi:10.1016/0097-3165(90)90006-i fatcat:wn2tgnkvx5hzfjzk2hbmukhgra

Inductive proofs of q-log concavity

Bruce E. Sagan
1992 Discrete Mathematics  
Note that while 6 is a total'order on N, s4 is only a partial order on N[q]. Still, this ordering respects the algebraic operations in N[q].  ...  The main result of [3] is that the aA(k;p) There are other possible candidates for the q-analogs of the definitions of q-log concavity and q-unimodal.  ...  This settles Butler's conjecture that the q-Stirling numbers of the second kind are strongly q-log concave when the second index varies.  ... 
doi:10.1016/0012-365x(92)90377-r fatcat:m2elcjl2bbc5zallrxdziuakiy

An overpartition analogue of q-binomial coefficients, II: combinatorial proofs and (q,t)-log concavity [article]

Jehanne Dousse, Byungchan Kim
2017 arXiv   pre-print
We particularly emphasize the role of combinatorial proofs and the consequences of our results on Delannoy numbers. We conclude with some conjectures about the unimodality of m+n n_q,t.  ...  Here, we add one more parameter counting the number of overlined parts, obtaining a two-parameter generalization m+n n_q,t of Gaussian polynomials, which is also a (q,t)-analogue of Delannoy numbers.  ...  By setting t = 0, we obtain Butler's result on the q-log-concavity of q-binomial coefficients.  ... 
arXiv:1701.07915v2 fatcat:6my2whnf3rhwne7x4misbszb64

Some applications of algebra to combinatorics

Richard P. Stanley
1991 Discrete Applied Mathematics  
based on my lectures.  ...  I am grateful to Rodica Simion for providing me the opportunity to lecture at the Capital City Conference on Combinatorics and Theoretical Computer Science, and for her careful preparation of these notes  ...  The lattice LA is rank unimodal (and rank symmetric) for all A. Butler's proof of rank unimodality uses the theory of symmetric functions; the rank symmetry is a well-known result.  ... 
doi:10.1016/0166-218x(91)90089-f fatcat:5qisuo24bzfiteuf6w6ryceb7u

Page 38 of Mathematical Reviews Vol. , Issue 2000j [page]

2000 Mathematical Reviews  
Burichenko (BE-AOS2; Gomel’) See also 57073, 57076. 20K Abelian groups 2000):20104 20K Abelian groups 2000j:20101 20K01 20K10 Takegahara, Yugen (J-MUIT; Muroran) On Butler’s unimodality result.  ...  One example of such a group is a direct product of cyclic groups of order p’.  ... 

The q-Log-Concavity and Unimodality of q-Kaplansky Numbers [article]

Kathy Q. Ji
2021 arXiv   pre-print
Based on the work of Reiner and Stanton, we establish the unimodality of q-Kaplansky numbers.  ...  Combining Lemma 3.1 and the unimodality of Gaussian polynomials, we obtain the following result. Lemma 3.2.  ...  We prove Lemma 3.1 by generalizing Butler's bijection [2] . To describe the proof, we need to recall some notation and terminology on partitions as in [1, Chapter 1] .  ... 
arXiv:2111.03369v1 fatcat:ew7wck3pfnhr5bgkre5kxw3gou

Distinctiveness in the face of gene flow: hybridization between specialist and generalist gartersnakes

BENJAMIN M. FITZPATRICK, JOHN S. PLACYK, JR, MATTHEW L. NIEMILLER, GARY S. CASPER, GORDON M. BURGHARDT
2008 Molecular Ecology  
Unimodal hybrid zones or hybrid swarms are composed predominantly of recombinant individuals whose genomes are patchworks of alleles derived from each parental lineage.  ...  Here we focus on three patterns of distinctiveness in the face of gene flow between gartersnake taxa in the Great Lakes region of North America.  ...  Numerical simulations show that k-means (with k = 2) will tend to split a bimodal distribution such that each group contains one of the modes, whereas a unimodal distribution will simply be split down  ... 
doi:10.1111/j.1365-294x.2008.03885.x pmid:18684137 fatcat:4q2bjuh7qnhxlewer4azo4hi54

ALFWorld: Aligning Text and Embodied Environments for Interactive Learning [article]

Mohit Shridhar, Xingdi Yuan, Marc-Alexandre Côté, Yonatan Bisk, Adam Trischler, Matthew Hausknecht
2021 arXiv   pre-print
BUTLER's simple, modular design factors the problem to allow researchers to focus on models for improving every piece of the pipeline (language understanding, planning, navigation, and visual scene understanding  ...  Trained on All Tasks with 50K episodes and evaluated in the embodied environment. Table 5 5 presents results for unimodal baseline comparisons to BUTLER.  ...  Results in with pre-recorded expert demonstrations causes a bigger performance drop of 30% on seen split of All Tasks.  ... 
arXiv:2010.03768v2 fatcat:bcj42swdcffjjlo4hll7pubsom

Generalized flags in finite Abelian p-groups

Lynne M. Butler
1991 Discrete Applied Mathematics  
Our main result shows this topological condition is equivalent to the simple requirement that p dominates A.  ...  We employ results obtained by Lascoux and Schiitzenberger in the theory of Hall-Littlewood symmetric functions.  ...  Regonati's new proof [14] that the lattice of subgroups of any finite Abelian pgroup is rank-unimodal is based on such a group-theoretic interpretation.  ... 
doi:10.1016/0166-218x(91)90079-c fatcat:xpttcppscne5tmjo4ysu5yzjke

Modeling foraging in the northern anchovy (Engraulis mordax): individual behavior can predict school dynamics and population biology

Peter Nonacs, Paul E. Smith, Marc Mangel
1998 Canadian Journal of Fisheries and Aquatic Sciences  
The availability of extensive information on bioenergetics of the northern anchovy (Engraulis mordax) allows us to use this species as a case study.  ...  Patterns from existing data sets on gut contents and growth rates from the field for juvenile E. mordax were consistent only with the prediction of maximizing survivorship.  ...  Gilboa, and an anonymous reviewer for many helpful comments on the manuscript. This work was supported by NSF grant OCE 90-16895 to M. Mangel. This is contribution 100 of the U.S.  ... 
doi:10.1139/cjfas-55-5-1179 fatcat:idw55zebynbzvh6p3ptqjwh7mm

Modeling foraging in the northern anchovy (Engraulis mordax): individual behavior can predict school dynamics and population biology

Peter Nonacs, Paul E Smith, Marc Mangel
1998 Canadian Journal of Fisheries and Aquatic Sciences  
The availability of extensive information on bioenergetics of the northern anchovy (Engraulis mordax) allows us to use this species as a case study.  ...  Patterns from existing data sets on gut contents and growth rates from the field for juvenile E. mordax were consistent only with the prediction of maximizing survivorship.  ...  Gilboa, and an anonymous reviewer for many helpful comments on the manuscript. This work was supported by NSF grant OCE 90-16895 to M. Mangel. This is contribution 100 of the U.S.  ... 
doi:10.1139/f98-010 fatcat:yb3rloypmreu5nk2kgofj5rpku

Page 488 of Mathematical Reviews Vol. 32, Issue Index [page]

Mathematical Reviews  
(English summary) 2000m:20088 — see also Corradi, Keresztély et al., 2000k:20070 Takegahara, Yugen On Butler’s unimodality result.  ...  Russian summary) (see 2000m:03014) Takegahara, Yugen On Butler’s unimodality result. (English summary) 2000j:20101 20K15 Torsion-free groups, finite rank Albrecht, Ulrich (with Goeters, H.  ... 

Representation through Legislative Redistricting: A Stochastic Model

Gary King
1989 American Journal of Political Science  
By this procedure, then, the model has one additional parameter.  ...  Nonpartisan redistricting processes substantially increase responsiveness but do not have as clear an effect on bias.  ...  One should not confuse the procedure in this section with Butler's (1951) method of deriving the seats-votes curve.  ... 
doi:10.2307/2111110 fatcat:ouqu6he6mfaihoqfdbjj6alzeq

Counting subspaces of a finite vector space [article]

Amritanshu Prasad
2011 arXiv   pre-print
In this expository article, we discuss the relation between the Gaussian binomial and multinomial coefficients and ordinary binomial and multinomial coefficients from a combinatorial viewpoint, based on  ...  The result is a pattern of * 's inside a grid with k rows and n − k columns.  ...  However, the analogues of (14) and (16) are more subtle, and can be found in Lynne Butler's beautiful monograph [2] .  ... 
arXiv:1006.2193v2 fatcat:kpqzulyv6fhbnenknih6rzjco4

How tumor growth can be influenced by delayed interactions between cancer cells and the microenvironment?

Dibakar Ghosh, Subhas Khajanchi, Sylvain Mangiarotti, Fabrice Denis, Syamal K. Dana, Christophe Letellier
2017 Biosystems (Amsterdam. Print)  
Hence, according to Butler's lemma [54] , the singular point E * remains stable for 205 τ 2 < τ 0 2 = 0.0956.  ...  − 0.8486z) − 0.703yz − 2.323xz (58) The topological properties of the resulting attractor were carefully compared with the original ones, and the values of parameter ρ in the second equation was adjusted  ... 
doi:10.1016/j.biosystems.2017.05.001 pmid:28506827 fatcat:kcjalikquvhlljh6mbwfctwnc4
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