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Partitions into parts simultaneously regular, distinct, and/or flat [article]

William J. Keith
<span title="2019-11-12">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We find results on their behavior and generating functions: more results for those simultaneously regular and distinct, fewest for those distinct and flat. We offer some conjectures in the area.  ...  We explore partitions that lie in the intersection of several sets of classical interest: partitions with parts indivisible by m, appearing fewer than m times, or differing by less than m.  ...  The first step φ t replaces parts of sizes jt k with appearances of the part j; since jt k was not divisible by s, neither is j, and so the result is also an s-regular, t-regular partition; all such partitions  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1911.04755v1">arXiv:1911.04755v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/om5jyvahgraobk5abufzasicga">fatcat:om5jyvahgraobk5abufzasicga</a> </span>
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An identity involving 3-regular graphs

D.R. Woodall
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
It is proved that, if M is a perfect matching in a 3-regular graph G, then the number of positive-minus-negative M-covers of G is equal to the number of positive-minus-negative M-partitions of G.  ...  Moreover, either there are no M-partitions of G, or every M-partition and every M-cover has the same sign.  ...  Also there are two M-partitions, ({{d,e}}, {{d,e'}}) and ({ {d, e'} }, {{d, e} }), and so P(H, M) = 2. Thus the result holds for both graphs with IMI = 1.Suppose now that IM I ;:: 2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(94)00274-m">doi:10.1016/0012-365x(94)00274-m</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ov2szw33qzbmhawzh4boy3whwe">fatcat:ov2szw33qzbmhawzh4boy3whwe</a> </span>
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4-Regular partitions and the pod function [article]

Cristina Ballantine, Mircea Merca
<span title="2021-11-21">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, for k∈{0,2} we consider the partitions of n into distinct parts not congruent to k modulo 4 and the 4-regular partitions of n in order to obtain new properties for pod(n).  ...  The partition function pod(n) enumerates the partitions of n wherein odd parts are distinct and even parts are unrestricted. Recently, a number of properties for pod(n) have been established.  ...  Thus, λe is a partition with distinct odd parts. Since λ o is a partition with distinct odd parts, we denote by λo the partition whose parts are the parts of ξ(λ o ) multiplied by 2.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2111.10702v1">arXiv:2111.10702v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/haha754skfhqbn544qxw3goszq">fatcat:haha754skfhqbn544qxw3goszq</a> </span>
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Sharp bounds for partition dimension of generalized Möbius ladders

Zafar Hussain, Junaid Alam Khan, Mobeen Munir, Muhammad Shoaib Saleem, Zaffar Iqbal
<span title="2018-11-08">2018</span> <i title="Walter de Gruyter GmbH"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/frczfdh6mjdyrnsaqiedlodjom" style="color: black;">Open Mathematics</a> </i> &nbsp;
In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, Mm, n, for all n≥3 and m≥2.  ...  It is hard to compute exact values of partition dimension for a graphic metric space, (G, dG) and networks.  ...  The generalized Möbius ladder obtained from P × P is: Main results and discussions In this part we give our main results.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1515/math-2018-0109">doi:10.1515/math-2018-0109</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/bd4rnap5jnhyjbthqnwpfn6lqi">fatcat:bd4rnap5jnhyjbthqnwpfn6lqi</a> </span>
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Arithmetic completely regular codes

Jacobus Koolen, Woo Sun Lee, William Martin, Hajime Tanaka
<span title="2016-02-27">2016</span> <i title="Centre pour la Communication Scientifique Directe (CCSD)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aagtqr2vajamvduhte7kigeygi" style="color: black;">Discrete Mathematics &amp; Theoretical Computer Science</a> </i> &nbsp;
Employing earlier results, we are then able to prove a theorem which nearly classifies these codes in the case where the graph admits a completely regular partition into such codes (e.g, the cosets of  ...  some additive completely regular code).  ...  Part of this work was completed while JHK was with the Pohang University of Science and Technology (POSTECH) and WJM was visiting him.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.46298/dmtcs.2150">doi:10.46298/dmtcs.2150</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/am67ah2sqrf75nclxsi5m33gnm">fatcat:am67ah2sqrf75nclxsi5m33gnm</a> </span>
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Translates of linear codes over Z/sub 4/

A. Bonnecaze, I.M. Duursma
<span title="">1997</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/niovmjummbcwdg4qshgzykkpfu" style="color: black;">IEEE Transactions on Information Theory</a> </i> &nbsp;
For the particular case of quaternary Preparata codes, we obtain that the number of distinct complete weights for the dual Preparata codes and the number of distinct complete coset weight enumerators for  ...  The method follows known ideas that have already been used successfully by others for Hamming weight distributions.  ...  ACKNOWLEDGMENT The authors wish to thank P. V. Kumar and P. Solé for their interest in this work and for stimulating discussions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/18.605585">doi:10.1109/18.605585</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qs2w4ubf4ffpfguvsimlz4k4q4">fatcat:qs2w4ubf4ffpfguvsimlz4k4q4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809221618/http://alexis.bonnecaze.perso.luminy.univ-amu.fr/articles/Translates1997.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/c3/ec/c3ecc760c9a2a79d85d19271a615d0d7da219081.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/18.605585"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> ieee.com </button> </a>

Parity and Partition of the Rational Numbers [article]

Peter Lynch, Michael Mackey
<span title="2022-05-01">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This pattern means that the three parity classes have equal natural density in the rationals. A similar result holds for the Stern-Brocot tree.  ...  Using the 2-adic valuation, we partition the rationals into subgroups with a rich algebraic structure. The natural density provides a means of distinguishing the sizes of countably infinite sets.  ...  University for guidance on p-adic numbers.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2205.00565v1">arXiv:2205.00565v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/j62p64qbxnh7zhikwjhx343364">fatcat:j62p64qbxnh7zhikwjhx343364</a> </span>
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Page 115 of Mathematical Reviews Vol. , Issue 2000a [page]

<span title="">2000</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
The author counts complete partitions with respect to their number of parts and size of largest part. He derives recurrences for the respective numbers.  ...  Using this fact the author studies g(m) [resp. go(n)], the number of partitions of an integer n into distinct [resp. distinct odd] parts.  ... 
<span class="external-identifiers"> </span>
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A special case of completion invariance for the c_2 invariant of a graph [article]

Karen Yeats
<span title="2018-01-25">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
This paper will prove completion invariance of the c_2 invariant in the case that p=2 and the completed graph has an odd number of vertices.  ...  It is useful for understanding Feynman periods. Brown and Schnetz conjectured that the c_2 invariant has a particular symmetry known as completion invariance.  ...  This last lemma is particularly useful when used in conjunction with the previous lemma. Suppose G is a decompletion of a 4-regular graph and i, j, k are distinct edge indices.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1706.08857v2">arXiv:1706.08857v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ecl3kspourfgrprtsqtprcxh4u">fatcat:ecl3kspourfgrprtsqtprcxh4u</a> </span>
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Algebraic characterizations of regularity properties in bipartite graphs [article]

A. Abiad, C. Dalfò, M.A. Fiol
<span title="2013-09-26">2013</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Moreover, it is shown that the conditions for having distance-regularity in such graphs can be relaxed when compared with general graphs.  ...  In this case, some characterizations of (bi)regular bipartite graphs are given in terms of the mean degrees in every partite set and the Hoffman polynomial.  ...  Moreover, if i and d have distinct parity, (p d (A)) uv = 0. Thus, we only need to consider the case when i(< d) and d have the same parity.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1205.5913v2">arXiv:1205.5913v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7p2y7rhzjbg67a4kxoritsr3j4">fatcat:7p2y7rhzjbg67a4kxoritsr3j4</a> </span>
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Indecomposable summands of Foulkes modules

Eugenio Giannelli, Mark Wildon
<span title="">2016</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/74pjbehpfraq5ogj3loetdwp7m" style="color: black;">Journal of Pure and Applied Algebra</a> </i> &nbsp;
of odd characteristic p.  ...  In this paper we study the modular structure of the permutation module H (2 n ) of the symmetric group S2n acting on set partitions of a set of size 2n into n sets each of size 2, defined over a field  ...  We note that P (10, 4) is the projective summand used in the proof of Theorem 1.1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jpaa.2016.01.013">doi:10.1016/j.jpaa.2016.01.013</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fkxt6eeiazhe7ivucr2hswouje">fatcat:fkxt6eeiazhe7ivucr2hswouje</a> </span>
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Binary codes with locality for multiple erasures having short block length

S. B. Balaji, K. P. Prasanth, P. Vijay Kumar
<span title="">2016</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ux5tf45dtrgptnezraiii73msm" style="color: black;">2016 IEEE International Symposium on Information Theory (ISIT)</a> </i> &nbsp;
In the case of sequential repair, the results include (a) extending and characterizing minimum-block-length constructions for t=2, (b) providing improved bounds on block length for t=3 as well as a general  ...  Both sequential and parallel (through orthogonal parity checks) recovery is considered here. In the case of parallel repair, minimum-block-length constructions for general t are discussed.  ...  Clearly, n 1 and n 2 belong to two different partitions, say A 1 and A 2 . Hyperedges i and j will be connected to two distinct nodes in the third partition A 3 .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1109/isit.2016.7541380">doi:10.1109/isit.2016.7541380</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/isit/BalajiPK16.html">dblp:conf/isit/BalajiPK16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/syr752ke2vbtxebtl2gthgfelm">fatcat:syr752ke2vbtxebtl2gthgfelm</a> </span>
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Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions

Chang-Tse Hsieh, Gil Young Cho, Shinsei Ryu
<span title="2016-02-18">2016</span> <i title="American Physical Society (APS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/b7jx7xdhxncvtj2nb24xhplinq" style="color: black;">Physical review B</a> </i> &nbsp;
For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under  ...  Similarly, for the second example, which is related to, by CPT-theorem, time-reversal symmetric topological superconductors, we discuss the invariance/non-invariance of the partition function of the surface  ...  GYC thanks deeply Yuan-Ming Lu for helpful discussion. SR is supported by Alfred P. Sloan Research Foundation and the NSF under Grant No. DMR-1455296.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1103/physrevb.93.075135">doi:10.1103/physrevb.93.075135</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/dhvg7xmmx5frtjkf2pktmjlfeu">fatcat:dhvg7xmmx5frtjkf2pktmjlfeu</a> </span>
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Edge-Connectivity, Eigenvalues, and Matchings in Regular Graphs

Suil O, Sebastian M. Cioabă
<span title="">2010</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/hn6wut4eynat5my6xdfsebg7fm" style="color: black;">SIAM Journal on Discrete Mathematics</a> </i> &nbsp;
We give a condition on an appropriate eigenvalue that guarantees a lower bound for the matching number of a t-edge-connected d-regular graph, when t ≤ d − 2.  ...  This work extends some classical results of von Baebler and Berge and more recent work of Cioabȃ, Gregory, and Haemers.  ...  West for his helpful comments and thank the anonymous referee for his/her remarks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/100786824">doi:10.1137/100786824</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/snpvnnzg6zbbtlulc3unttejy4">fatcat:snpvnnzg6zbbtlulc3unttejy4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170830070847/http://www.math.udel.edu/~cioaba/PaperOSept10.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/82/23/82239e45b40bb4efdc5f4384a33dba15ed08233e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/100786824"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Genomic Selection for Adjacent Genetic Markers of Yorkshire Pigs Using Regularized Regression Approaches

Minsu Park, Tae-Hun Kim, Eun-Seok Cho, Heebal Kim, Hee-Seok Oh
<span title="2014-10-16">2014</span> <i title="Asian Australasian Association of Animal Production Societies"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/5x4vaquupnaarlclnfczoqwwd4" style="color: black;">Asian-Australasian Journal of Animal Sciences</a> </i> &nbsp;
For efficient selection of variables with specific traits in pig breeding, it is required that any such variable selection retains some properties: i) it produces a simple model by identifying insignificant  ...  and litter size for 519 observed sows.  ...    P   where the penalty function is 1 ) ( l P      with a regularization parameter λ≥0.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5713/ajas.2014.14236">doi:10.5713/ajas.2014.14236</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/25358359">pmid:25358359</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC4213677/">pmcid:PMC4213677</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kohdqgcconc4lpegci6ttwtsvm">fatcat:kohdqgcconc4lpegci6ttwtsvm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170814101930/http://ocean.kisti.re.kr/downfile/volume/aaaaps/E1DMBP/2014/v27n12/E1DMBP_2014_v27n12_1678.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6f/e1/6fe1b27c0f26fdbf9c4ca959d238dd271a02be25.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.5713/ajas.2014.14236"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4213677" title="pubmed link"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> pubmed.gov </button> </a>
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