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Total positivity of Narayana matrices [article]

Yi Wang, Arthur L.B. Yang
2017 arXiv   pre-print
We also prove the strict total positivity of the Narayana squares of type A and type B.  ...  We prove the total positivity of the Narayana triangles of type A and type B, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng.  ...  Acknowledgements This work was supported by the National Science Foundation of China (Nos. 11231004, 11371078, 11522110) .  ... 
arXiv:1702.07822v1 fatcat:m3ndhgblrbeerk4tlizzpwucdm

Total positivity of recursive matrices

Xi Chen, Huyile Liang, Yi Wang
2015 Linear Algebra and its Applications  
As applications we give the total positivity of the above mentioned combinatorial triangles in a unified approach.  ...  We present some sufficient conditions such that the recursive matrix A is totally positive.  ...  The motivation of this paper is to study the total positivity of these combinatorial triangles. Corollary 2.10.  ... 
doi:10.1016/j.laa.2015.01.009 fatcat:rrhxsfavo5cn7pqzuliw2zfu7a

Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials [article]

Nikita Alexeev, Alexander Tikhomirov
2016 arXiv   pre-print
We consider Gaussian elliptic random matrices X of a size N × N with parameter ρ, i.e., matrices whose pairs of entries (X_ij, X_ji) are mutually independent Gaussian vectors, E X_ij = 0, E X^2_ij = 1  ...  We have proven that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B: c_2n = ∑_k=0^n nk^2 ρ^2k.  ...  The work of NA is supported by the grant of Russian Scientific Foundation 14-11-00581. The work of AT is supported by SFB 701 Spectral Structures  ... 
arXiv:1501.04615v3 fatcat:mxp3iqrfrvczbhhqkqfgvlqk2e

Log-convex and Stieltjes moment sequences [article]

Yi Wang, Bao-Xuan Zhu
2016 arXiv   pre-print
the infinite log-convexity of the Schröder numbers.  ...  We introduce the concept of q-Stieltjes moment sequences of polynomials and show that many well-known polynomials in combinatorics are such sequences.  ...  Acknowledgements This work was supported in part by the National Natural Science Foundation of China (Nos. 11201191, 11371078, 11571150) .  ... 
arXiv:1612.04114v1 fatcat:4kdv75hd6jhevo5grledspbdtq

Relative Entropy of Random States and Black Holes [article]

Jonah Kudler-Flam
2021 arXiv   pre-print
Finally, we interpret these results in the context of the subsystem Eigenstate Thermalization Hypothesis (sETH), concluding that holographic systems obey sETH up to subsystems half the size of the total  ...  Furthermore, the random matrix theory results accurately match the behavior of chaotic many-body eigenstates, a manifestation of eigenstate thermalization.  ...  constant (G N ) up until the subregion is half of the total system size.  ... 
arXiv:2102.05053v2 fatcat:j5ubhqwy2fecpmux4ivnpdxg3e

Generalizing Narayana and Schröder Numbers to Higher Dimensions

Robert A. Sulanke
2004 Electronic Journal of Combinatorics  
Define the generalized Narayana number $N(d,n,k)$ to count the paths in ${\cal C}(d,n)$ with ${\rm asc}(P)=k$. We consider the derivation of a formula for $N(d,n,k)$, implicit in MacMahon's work.  ...  Let ${\cal C}(d,n)$ denote the set of $d$-dimensional lattice paths using the steps $X_1 := (1, 0, \ldots, 0),$ $ X_2 := (0, 1, \ldots, 0),$ $\ldots,$ $ X_d := (0,0, \ldots,1)$, running from $(0,\ldots  ...  He thanks the referee for several positive suggestions including a comment which led to the conjecture at the end of the remarks of §3.1.  ... 
doi:10.37236/1807 fatcat:tpvw5cbyqjfbheehmwzmbodz2a

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes–Rogers and Thron–Rogers polynomials, with coefficientwise Hankel-total positivity [article]

Mathias Pétréolle, Alan D. Sokal, Bao-Xuan Zhu
2020 arXiv   pre-print
s=0 we prove the coefficientwise Hankel-total positivity.  ...  We prove that all of these sequences of polynomials are coefficientwise Hankel-totally positive, jointly in all the (infinitely many) indeterminates.  ...  We also wish to thank the organizers of the 80th Séminaire Lotharingien de Combinatoire (Lyon, March 2018) for the opportunity to present a preliminary version of this work.  ... 
arXiv:1807.03271v2 fatcat:mhzpgcz5fzawxk5uzp72imemse

Three dimensional Narayana and Schröder numbers

Robert A. Sulanke
2005 Theoretical Computer Science  
We define other statistics distributed by the 3-Narayana number and show that 4 k 2 k N(3, n, k) yields the nth large 3-Schröder number which counts the constrained paths using the seven positive steps  ...  A combinatorial cancellation argument and a result of MacMahon yield a formula for the 3-Narayana number, N(3, n, k) := |{P ∈ C(3, n) : des(P ) = k + 2}|.  ...  Finally, he is grateful to the referees for their positive suggestions.  ... 
doi:10.1016/j.tcs.2005.08.014 fatcat:ghhasjdfnzad3adbv4qpam4u5i

Page 1231 of Mathematical Reviews Vol. , Issue 88c [page]

1988 Mathematical Reviews  
Let k and N be positive integers and let y be a partition of N.  ...  M. (1-NM) Enumerating 3 x 3 N-matrices with given row and column sums.  ... 

Page 3259 of Mathematical Reviews Vol. , Issue 96f [page]

1996 Mathematical Reviews  
In the first part of his paper, the author proves a combinatorial characterization of totally positive matrices.  ...  Totally positive matrices are important in various branches of mathematics, statis- tics, mechanics, economics and computer science.  ... 

Extensions of partial cyclic orders and consecutive coordinate polytopes [article]

Arvind Ayyer and Matthieu Josuat-Vergès and Sanjay Ramassamy
2019 arXiv   pre-print
We also provide a combinatorial interpretation of the Ehrhart h^*-polynomials of some of these polytopes in terms of descents of total cyclic orders.  ...  The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.  ...  Such matrices are called interval matrices and form a well-known class of totally unimodular matrices [Sch86, Chapter 19, Example 7]. Lemma 8.  ... 
arXiv:1803.10351v3 fatcat:lrcic7iczfaevlqwjvxu3lg2ve

Extensions of partial cyclic orders and consecutive coordinate polytopes

Arvind Ayyer, Matthieu Josuat-Vergès, Sanjay Ramassamy
2020 Annales Henri Lebesgue  
We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders.  ...  We introduce several classes of polytopes contained in [0, 1] n and cut out by inequalities involving sums of consecutive coordinates.  ...  Such matrices are called interval matrices and form a well-known class of totally unimodular matrices [Sch86, Chapter 19, Example 7]. Lemma 3.2.  ... 
doi:10.5802/ahl.33 fatcat:rfbw52ctcnd2hegyelpbqusese

Positivity of iterated sequences of polynomials [article]

Bao-Xuan Zhu
2018 arXiv   pre-print
, the Eulerian polynomials of Types A and B, the q-Schröder numbers, q-central Delannoy numbers, the Narayana polynomials of Types A and B, the generating functions of rows in the Catalan triangles of  ...  These allow a unified treatment of the 2-q-log-convexity of alternating Eulerian polynomials, 2-log-convexity of Euler numbers, and 3-q-log-convexity of many classical polynomials, including the Bell polynomials  ...  In addition, results of this paper were presented in the Institute of Mathematics, Academia Sinica, Taipei in Jan. 2016, the seventh National Conference on Combinatorics and Graph Theory at Hebei Normal  ... 
arXiv:1807.01062v1 fatcat:qv2tlw5m4zeh5jkrd24xf3dsti

Methods to Determine Vertical Jaw Relation- Review

Dr. Priscilla Shalini, Narayana Reddy
2020 American Journal of PharmTech Research  
The study of device in terms of composition and physical principles, technical such as the use of one-dimensional ultrasound.  ...  the number of cases for each month from January to July 2019.Statistics were created then represented on the graph by combining Excel and Word.  ...  Most imaging systems provide image sizes of (256,256), (512, 512), and (1024, 1024) matrices, image resolution will be better with a larger image matrix.  ... 
doi:10.46624/ajptr.2020.v10.i3.002 fatcat:yayww6kg4bdppongpczhkx7alq

Yang-Baxter Integrable Dimers on a Strip [article]

Paul A. Pearce, Jørgen Rasmussen, Alessandra Vittorini-Orgeas
2019 arXiv   pre-print
We further show that the corresponding finitized characters _s^(N)(q) decompose into sums of q-Narayana numbers or, equivalently, skew q-binomials.  ...  In the particle representation, the local face tile operators give a representation of the fermion algebra and the fermion particle trajectories play the role of nonlocal degrees of freedom.  ...  For a column at position j with fixed a j , b j = 0, 1, let us define the four 16 × 16 matrices elements of the product of double row transfer matrices, with upper and lower particle state configurations  ... 
arXiv:1907.07610v2 fatcat:gqwuqfvatrg3nbfpkhdkqqqhhq
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