• Previous Article
    Global dynamics in a competitive two-species and two-stimuli chemotaxis system with chemical signalling loop
  • ERA Home
  • This Issue
  • Next Article
    Multiple-site deep brain stimulation with delayed rectangular waveforms for Parkinson's disease
doi: 10.3934/era.2021018
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Refined Wilf-equivalences by Comtet statistics

1. 

College of Mathematics and Statistics, Chongqing University, Huxi campus, Chongqing 401331, China

2. 

Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, China

* Corresponding author: Shishuo Fu

Received  May 2020 Revised  November 2020 Early access March 2021

Fund Project: The second author was supported by the National Science Foundation of China grants 11871247 and the project of Qilu Young Scholars of Shandong University.

We launch a systematic study of the refined Wilf-equivalences by the statistics $ {\mathsf{comp}} $ and $ {\mathsf{iar}} $, where $ {\mathsf{comp}}(\pi) $ and $ {\mathsf{iar}}(\pi) $ are the number of components and the length of the initial ascending run of a permutation $ \pi $, respectively. As Comtet was the first one to consider the statistic $ {\mathsf{comp}} $ in his book Analyse combinatoire, any statistic equidistributed with $ {\mathsf{comp}} $ over a class of permutations is called by us a Comtet statistic over such class. This work is motivated by a triple equidistribution result of Rubey on $ 321 $-avoiding permutations, and a recent result of the first and third authors that $ {\mathsf{iar}} $ is a Comtet statistic over separable permutations. Some highlights of our results are:

● Bijective proofs of the symmetry of the joint distribution $ ({\mathsf{comp}}, {\mathsf{iar}}) $ over several Catalan and Schröder classes, preserving the values of the left-to-right maxima.

● A complete classification of $ {\mathsf{comp}} $- and $ {\mathsf{iar}} $-Wilf-equivalences for length $ 3 $ patterns and pairs of length $ 3 $ patterns. Calculations of the $ ({\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}) $ generating functions over these pattern avoiding classes and separable permutations.

● A further refinement of Wang's descent-double descent-Wilf equivalence between separable permutations and $ (2413, 4213) $-avoiding permutations by the Comtet statistic $ {\mathsf{iar}} $.

Citation: Shishuo Fu, Zhicong Lin, Yaling Wang. Refined Wilf-equivalences by Comtet statistics. Electronic Research Archive, doi: 10.3934/era.2021018
References:
[1]

R. M. AdinE. Bagno and Y. Roichman, Block decomposition of permutations and Schur-positivity, J. Algebraic Combin., 47 (2018), 603-622.  doi: 10.1007/s10801-017-0788-9.  Google Scholar

[2]

C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2018), Article B77i, 64 pp (electronic).  Google Scholar

[3]

M. Barnabei, F. Bonetti and M. Silimbani, The descent statistic on $123$-avoiding permutations, Sém. Lothar. Combin., 63 (2010), B63a, 8 pp.  Google Scholar

[4]

S. C. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for $321$-Hexagon-avoiding permutations, J. Algebraic Combin., 13 (2001), 111-136.  doi: 10.1023/A:1011279130416.  Google Scholar

[5]

M. Bóna, Combinatorics of Permutations. With a Foreword by Richard Stanley, Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203494370.  Google Scholar

[6]

F. R. K. ChungR. L. GrahamV. E. Hoggatt Jr. and M. Kleiman, The number of Baxter permutations, J. Combin. Theory Ser. A, 24 (1978), 382-394.  doi: 10.1016/0097-3165(78)90068-7.  Google Scholar

[7]

A. Claesson and S. Kitaev, Classification of bijections between $321$-and $132$-avoiding permutations, Sém. Lothar. Combin., 60 (2008), B60d, 30 pp.  Google Scholar

[8]

A. ClaessonS. Kitaev and E. Steingrímsson, Decompositions and statistics for $\beta(1,0)$-trees and nonseparable permutations, Adv. in Appl. Math., 42 (2009), 313-328.  doi: 10.1016/j.aam.2008.09.001.  Google Scholar

[9]

L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.  Google Scholar

[10]

S. Corteel, M. A. Martinez, C. D. Savage and M. Weselcouch, Patterns in inversion sequences I, Discrete Math. Theor. Comput. Sci., 18 (2016), Paper No. 2, 21 pp.  Google Scholar

[11]

T. DokosT. DwyerB. P. JohnsonB. E. Sagan and K. Selsor, Permutation patterns and statistics, Discrete Math., 312 (2012), 2760-2775.  doi: 10.1016/j.disc.2012.05.014.  Google Scholar

[12]

P. G. Doyle, Stackable and queueable permutations, preprint, arXiv: 1201.6580. Google Scholar

[13]

S. Elizalde and I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A, 105 (2004), 207-219.  doi: 10.1016/j.jcta.2003.10.009.  Google Scholar

[14]

D. Foata and M.-P. Schützenberger, Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, Vol. 138, Springer-Verlag, Berlin-New York, 1970.  Google Scholar

[15]

S. FuG. -N. Han and Z. Lin, $k$-arrangements, statistics, and patterns, SIAM J. Discrete Math., 34 (2020), 1830-1853.  doi: 10.1137/20M1340538.  Google Scholar

[16]

S. Fu, Z. Lin and Y. Wang, A combinatorial bijection on di-sk trees, preprint, arXiv: 2011.11302. Google Scholar

[17]

S. FuZ. Lin and J. Zeng, On two new unimodal descent polynomials, Discrete Math., 341 (2018), 2616-2626.  doi: 10.1016/j.disc.2018.06.010.  Google Scholar

[18]

S. Fu and Y. Wang, Bijective proofs of recurrences involving two Schröder triangles, European J. Combin., 86 (2020), 103077, 18 pp. doi: 10.1016/j.ejc.2019.103077.  Google Scholar

[19]

A. L. L. Gao, S. Kitaev and P. B. Zhang, On pattern avoiding indecomposable permutations, Integers, 18 (2018), A2, 23 pp.  Google Scholar

[20]

S. Kitaev, Patterns in Permutations and Words. With a Forewrod by Jeffrey B. Remmel, Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-17333-2.  Google Scholar

[21]

D. E. Knuth, The Art of Computer Programming. Vol. 1. Fundamental Algorithms, 3$^{rd}$ edition, Addison-Wesley, Reading, MA, 1997.  Google Scholar

[22]

C. Krattenthaler, Permutations with restricted patterns and Dyck paths, Special Issue in Honor of Dominique Foata's 65th birthday, Adv. in Appl. Math., 27 (2001), 510-530.  doi: 10.1006/aama.2001.0747.  Google Scholar

[23]

Z. Lin, On $\gamma$-positive polynomials arising in pattern avoidance, Adv. in Appl. Math., 82 (2017), 1-22.  doi: 10.1016/j.aam.2016.06.001.  Google Scholar

[24]

Z. Lin and D. Kim, A sextuple equidistribution arising in pattern avoidance, J. Combin. Theory Ser. A, 155 (2018), 267-286.  doi: 10.1016/j.jcta.2017.11.009.  Google Scholar

[25]

OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2020. Google Scholar

[26]

T. K. Petersen, Eulerian numbers. With a Foreword by Richard Stanley, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer, New York, 2015. doi: 10.1007/978-1-4939-3091-3.  Google Scholar

[27]

M. Rubey, An involution on Dyck paths that preserves the rise composition and interchanges the number of returns and the position of the first double fall, Sém. Lothar. Combin., 77 (2016-2018), Art. B77f, 4 pp.  Google Scholar

[28]

R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6 (1985), 383-406.  doi: 10.1016/S0195-6698(85)80052-4.  Google Scholar

[29]

Z. E. Stankova, Forbidden subsequences, Discrete Math., 132 (1994), 291-316.  doi: 10.1016/0012-365X(94)90242-9.  Google Scholar

[30]

D. Wang, The Eulerian distribution on involutions is indeed $\gamma$-positive, J. Combin. Theory Ser. A, 165 (2019), 139-151.  doi: 10.1016/j.jcta.2019.02.004.  Google Scholar

[31]

J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math., 146 (1995), 247-262.  doi: 10.1016/0012-365X(94)00067-1.  Google Scholar

show all references

References:
[1]

R. M. AdinE. Bagno and Y. Roichman, Block decomposition of permutations and Schur-positivity, J. Algebraic Combin., 47 (2018), 603-622.  doi: 10.1007/s10801-017-0788-9.  Google Scholar

[2]

C. A. Athanasiadis, Gamma-positivity in combinatorics and geometry, Sém. Lothar. Combin., 77 (2018), Article B77i, 64 pp (electronic).  Google Scholar

[3]

M. Barnabei, F. Bonetti and M. Silimbani, The descent statistic on $123$-avoiding permutations, Sém. Lothar. Combin., 63 (2010), B63a, 8 pp.  Google Scholar

[4]

S. C. Billey and G. S. Warrington, Kazhdan-Lusztig polynomials for $321$-Hexagon-avoiding permutations, J. Algebraic Combin., 13 (2001), 111-136.  doi: 10.1023/A:1011279130416.  Google Scholar

[5]

M. Bóna, Combinatorics of Permutations. With a Foreword by Richard Stanley, Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2004. doi: 10.1201/9780203494370.  Google Scholar

[6]

F. R. K. ChungR. L. GrahamV. E. Hoggatt Jr. and M. Kleiman, The number of Baxter permutations, J. Combin. Theory Ser. A, 24 (1978), 382-394.  doi: 10.1016/0097-3165(78)90068-7.  Google Scholar

[7]

A. Claesson and S. Kitaev, Classification of bijections between $321$-and $132$-avoiding permutations, Sém. Lothar. Combin., 60 (2008), B60d, 30 pp.  Google Scholar

[8]

A. ClaessonS. Kitaev and E. Steingrímsson, Decompositions and statistics for $\beta(1,0)$-trees and nonseparable permutations, Adv. in Appl. Math., 42 (2009), 313-328.  doi: 10.1016/j.aam.2008.09.001.  Google Scholar

[9]

L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974.  Google Scholar

[10]

S. Corteel, M. A. Martinez, C. D. Savage and M. Weselcouch, Patterns in inversion sequences I, Discrete Math. Theor. Comput. Sci., 18 (2016), Paper No. 2, 21 pp.  Google Scholar

[11]

T. DokosT. DwyerB. P. JohnsonB. E. Sagan and K. Selsor, Permutation patterns and statistics, Discrete Math., 312 (2012), 2760-2775.  doi: 10.1016/j.disc.2012.05.014.  Google Scholar

[12]

P. G. Doyle, Stackable and queueable permutations, preprint, arXiv: 1201.6580. Google Scholar

[13]

S. Elizalde and I. Pak, Bijections for refined restricted permutations, J. Combin. Theory Ser. A, 105 (2004), 207-219.  doi: 10.1016/j.jcta.2003.10.009.  Google Scholar

[14]

D. Foata and M.-P. Schützenberger, Théorie Géométrique des Polynômes Eulériens, Lecture Notes in Mathematics, Vol. 138, Springer-Verlag, Berlin-New York, 1970.  Google Scholar

[15]

S. FuG. -N. Han and Z. Lin, $k$-arrangements, statistics, and patterns, SIAM J. Discrete Math., 34 (2020), 1830-1853.  doi: 10.1137/20M1340538.  Google Scholar

[16]

S. Fu, Z. Lin and Y. Wang, A combinatorial bijection on di-sk trees, preprint, arXiv: 2011.11302. Google Scholar

[17]

S. FuZ. Lin and J. Zeng, On two new unimodal descent polynomials, Discrete Math., 341 (2018), 2616-2626.  doi: 10.1016/j.disc.2018.06.010.  Google Scholar

[18]

S. Fu and Y. Wang, Bijective proofs of recurrences involving two Schröder triangles, European J. Combin., 86 (2020), 103077, 18 pp. doi: 10.1016/j.ejc.2019.103077.  Google Scholar

[19]

A. L. L. Gao, S. Kitaev and P. B. Zhang, On pattern avoiding indecomposable permutations, Integers, 18 (2018), A2, 23 pp.  Google Scholar

[20]

S. Kitaev, Patterns in Permutations and Words. With a Forewrod by Jeffrey B. Remmel, Monographs in Theoretical Computer Science. An EATCS Series. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-17333-2.  Google Scholar

[21]

D. E. Knuth, The Art of Computer Programming. Vol. 1. Fundamental Algorithms, 3$^{rd}$ edition, Addison-Wesley, Reading, MA, 1997.  Google Scholar

[22]

C. Krattenthaler, Permutations with restricted patterns and Dyck paths, Special Issue in Honor of Dominique Foata's 65th birthday, Adv. in Appl. Math., 27 (2001), 510-530.  doi: 10.1006/aama.2001.0747.  Google Scholar

[23]

Z. Lin, On $\gamma$-positive polynomials arising in pattern avoidance, Adv. in Appl. Math., 82 (2017), 1-22.  doi: 10.1016/j.aam.2016.06.001.  Google Scholar

[24]

Z. Lin and D. Kim, A sextuple equidistribution arising in pattern avoidance, J. Combin. Theory Ser. A, 155 (2018), 267-286.  doi: 10.1016/j.jcta.2017.11.009.  Google Scholar

[25]

OEIS Foundation Inc., The On-Line Encyclopedia of Integer Sequences, http://oeis.org, 2020. Google Scholar

[26]

T. K. Petersen, Eulerian numbers. With a Foreword by Richard Stanley, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer, New York, 2015. doi: 10.1007/978-1-4939-3091-3.  Google Scholar

[27]

M. Rubey, An involution on Dyck paths that preserves the rise composition and interchanges the number of returns and the position of the first double fall, Sém. Lothar. Combin., 77 (2016-2018), Art. B77f, 4 pp.  Google Scholar

[28]

R. Simion and F. W. Schmidt, Restricted permutations, European J. Combin., 6 (1985), 383-406.  doi: 10.1016/S0195-6698(85)80052-4.  Google Scholar

[29]

Z. E. Stankova, Forbidden subsequences, Discrete Math., 132 (1994), 291-316.  doi: 10.1016/0012-365X(94)90242-9.  Google Scholar

[30]

D. Wang, The Eulerian distribution on involutions is indeed $\gamma$-positive, J. Combin. Theory Ser. A, 165 (2019), 139-151.  doi: 10.1016/j.jcta.2019.02.004.  Google Scholar

[31]

J. West, Generating trees and the Catalan and Schröder numbers, Discrete Math., 146 (1995), 247-262.  doi: 10.1016/0012-365X(94)00067-1.  Google Scholar

Figure 1.  First three levels of the generating tree for $ \cup_{n\geq1}{\mathfrak{S}}_n(2431, 4231) $
Figure 2.  The block decomposition of separable permutations
Table 1.  One pattern of length $ 3 $ (definitions of $ N $, $ C $ and $ C^* $ are given in equations (12), (18) and (28), respectively)
$ P $ $ \tilde{\mathfrak{S}}^{{\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}}(t, r, p) $ $ M_n(P;{\mathsf{iar}}, {\mathsf{comp}}) $ proved in
$ 312 $ $ \dfrac{1-(r+p+tN)z+(rp+(r+p-1)tN)z^2}{(1-rpz)(1-rz-tNz)(1-pz-tNz)} $ Symmetric Thm. 3.9
$ 321 $ $ \dfrac{(rpz-rz+tz)C^2-(rpz+p-1)C+p}{(1-rpz)(1-rzC)(p+C-pC)} $ $ M_n(312) $ Thm. 3.10
$ 132 $ $ \dfrac{1}{1-rpz}+\dfrac{(1-z)(N-1)t}{(1-rz)(1-pz)(1-z-(N-1)tz)} $ Hankel Thm. 3.13
$ 213 $ $ \dfrac{(1-rz)(tN-t+1)}{(1-rpz)(1-rz(tN-t+1))} $ Lower triangular Thm. 3.15
$ 231 $ $ \dfrac{(1-pz)(tN-t+1)}{(1-rpz)(1-pz(tN-t+1))} $ $ M_n(213)^T $ Thm. 3.15
$ 123 $ $ \dfrac{(1-p)z(trz-tz-r)}{(1-tz)^2}+\dfrac{(1+rz-tz) C^*}{z(1+z-tz)} $ $ 2\times 2 $ nonzero Thm. 3.16
$ P $ $ \tilde{\mathfrak{S}}^{{\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}}(t, r, p) $ $ M_n(P;{\mathsf{iar}}, {\mathsf{comp}}) $ proved in
$ 312 $ $ \dfrac{1-(r+p+tN)z+(rp+(r+p-1)tN)z^2}{(1-rpz)(1-rz-tNz)(1-pz-tNz)} $ Symmetric Thm. 3.9
$ 321 $ $ \dfrac{(rpz-rz+tz)C^2-(rpz+p-1)C+p}{(1-rpz)(1-rzC)(p+C-pC)} $ $ M_n(312) $ Thm. 3.10
$ 132 $ $ \dfrac{1}{1-rpz}+\dfrac{(1-z)(N-1)t}{(1-rz)(1-pz)(1-z-(N-1)tz)} $ Hankel Thm. 3.13
$ 213 $ $ \dfrac{(1-rz)(tN-t+1)}{(1-rpz)(1-rz(tN-t+1))} $ Lower triangular Thm. 3.15
$ 231 $ $ \dfrac{(1-pz)(tN-t+1)}{(1-rpz)(1-pz(tN-t+1))} $ $ M_n(213)^T $ Thm. 3.15
$ 123 $ $ \dfrac{(1-p)z(trz-tz-r)}{(1-tz)^2}+\dfrac{(1+rz-tz) C^*}{z(1+z-tz)} $ $ 2\times 2 $ nonzero Thm. 3.16
Table 2.  Two patterns of length $ 3 $
$ P=(\tau_1, \tau_2) $ $ \tilde {\mathfrak{S}}^{{\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}}(t, r, p) $ $ M_n(P;{\mathsf{iar}}, {\mathsf{comp}}) $ proved in
$ (132, 312) $ $ \dfrac{1}{1-rpz}+\dfrac{(1-z)tz}{(1-rz)(1-pz)(1-z-tz)} $ Hankel Thm. 4.2
$ (132, 321) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-rz)(1-pz)(1-z)} $ $ 0 $-$ 1 $ Hankel Thm. 4.2
$ (213, 231) $ $ \dfrac{1-z}{(1-rpz)(1-z-tz)} $ Diagonal Thm. 4.3
$ (123, 312) $ $ \dfrac{1+rpz}{1-tz}+\dfrac{(r+p)tz^2}{(1-tz)^2}+\dfrac{t^2z^3}{(1-tz)^3} $ $ 2\times2 $ Hankel Thm. 4.4
$ (213, 312) $ $ \dfrac{1-rz}{(1-rpz)(1-(r+t)z)} $ Lower triangular Thm. 4.6
$ (231, 312) $ $ \dfrac{1-pz}{(1-rpz)(1-(p+t)z)} $ $ M_n(213, 312)^{T} $ Thm. 4.6
$ (231, 321) $ $ \dfrac{1-(1+p-t)z+(1-t)pz^2}{(1-rpz)(1-(p+1)z+(1-t)pz^2)} $ Upper triangular Thm. 4.6
$ (132, 213) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-rz)(1-z-tz)} $ Lower triangular Thm. 4.8
$ (132, 231) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-pz)(1-z-tz)} $ $ M_n(132, 213)^{T} $ Thm. 4.8
$ (213, 321) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-z)(1-rz)(1-rpz)} $ Lower triangular Thm. 4.9
$ (312, 321) $ $ \frac{1}{1-rpz}+\frac{(1-z)tz}{(1-rpz)(1-rz)(1-(1+p)z+(1-t)pz^2)} $ No pattern Thm. 4.10
$ (123, 132) $ $ 1+rpz+\frac{tpz^2}{1-tz}+\frac{tz(1+z-tz)(1+(r-t)z+(1-r)tz^2)}{(1-tz)((1-tz)^2-tz^2)} $ $ 2\times 2 $ nonzero Thm. 4.11
$ (123, 213) $ $ 1+\dfrac{rpz}{1-tz}+\dfrac{tz(1-tz+rz)(1-tz+z)}{(1-tz)((1-tz)^2-tz^2)} $ $ 2\times 2 $ nonzero Thm. 4.11
$ (123, 231) $ $ \dfrac{1+rpz}{1-tz}+\dfrac{(1+p-tpz)tz^2}{(1-tz)^3} $ $ 2\times 2 $ nonzero Thm. 4.11
$ (123, 321) $ $ \begin{array}{c} 1+(t+rp)z+(1+r)(1+p)tz^2\\ +(2r+t+pt)tz^3 \end{array} $ Ultimately zero Thm. 4.11
$ P=(\tau_1, \tau_2) $ $ \tilde {\mathfrak{S}}^{{\mathsf{des}}, {\mathsf{iar}}, {\mathsf{comp}}}(t, r, p) $ $ M_n(P;{\mathsf{iar}}, {\mathsf{comp}}) $ proved in
$ (132, 312) $ $ \dfrac{1}{1-rpz}+\dfrac{(1-z)tz}{(1-rz)(1-pz)(1-z-tz)} $ Hankel Thm. 4.2
$ (132, 321) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-rz)(1-pz)(1-z)} $ $ 0 $-$ 1 $ Hankel Thm. 4.2
$ (213, 231) $ $ \dfrac{1-z}{(1-rpz)(1-z-tz)} $ Diagonal Thm. 4.3
$ (123, 312) $ $ \dfrac{1+rpz}{1-tz}+\dfrac{(r+p)tz^2}{(1-tz)^2}+\dfrac{t^2z^3}{(1-tz)^3} $ $ 2\times2 $ Hankel Thm. 4.4
$ (213, 312) $ $ \dfrac{1-rz}{(1-rpz)(1-(r+t)z)} $ Lower triangular Thm. 4.6
$ (231, 312) $ $ \dfrac{1-pz}{(1-rpz)(1-(p+t)z)} $ $ M_n(213, 312)^{T} $ Thm. 4.6
$ (231, 321) $ $ \dfrac{1-(1+p-t)z+(1-t)pz^2}{(1-rpz)(1-(p+1)z+(1-t)pz^2)} $ Upper triangular Thm. 4.6
$ (132, 213) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-rz)(1-z-tz)} $ Lower triangular Thm. 4.8
$ (132, 231) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-pz)(1-z-tz)} $ $ M_n(132, 213)^{T} $ Thm. 4.8
$ (213, 321) $ $ \dfrac{1}{1-rpz}+\dfrac{tz}{(1-z)(1-rz)(1-rpz)} $ Lower triangular Thm. 4.9
$ (312, 321) $ $ \frac{1}{1-rpz}+\frac{(1-z)tz}{(1-rpz)(1-rz)(1-(1+p)z+(1-t)pz^2)} $ No pattern Thm. 4.10
$ (123, 132) $ $ 1+rpz+\frac{tpz^2}{1-tz}+\frac{tz(1+z-tz)(1+(r-t)z+(1-r)tz^2)}{(1-tz)((1-tz)^2-tz^2)} $ $ 2\times 2 $ nonzero Thm. 4.11
$ (123, 213) $ $ 1+\dfrac{rpz}{1-tz}+\dfrac{tz(1-tz+rz)(1-tz+z)}{(1-tz)((1-tz)^2-tz^2)} $ $ 2\times 2 $ nonzero Thm. 4.11
$ (123, 231) $ $ \dfrac{1+rpz}{1-tz}+\dfrac{(1+p-tpz)tz^2}{(1-tz)^3} $ $ 2\times 2 $ nonzero Thm. 4.11
$ (123, 321) $ $ \begin{array}{c} 1+(t+rp)z+(1+r)(1+p)tz^2\\ +(2r+t+pt)tz^3 \end{array} $ Ultimately zero Thm. 4.11
[1]

Ji-Cai Liu. Proof of Sun's conjectural supercongruence involving Catalan numbers. Electronic Research Archive, 2020, 28 (2) : 1023-1030. doi: 10.3934/era.2020054

[2]

Harald Friedrich. Semiclassical and large quantum number limits of the Schrödinger equation. Conference Publications, 2003, 2003 (Special) : 288-294. doi: 10.3934/proc.2003.2003.288

[3]

Michel Laurent, Arnaldo Nogueira. Rotation number of contracted rotations. Journal of Modern Dynamics, 2018, 12: 175-191. doi: 10.3934/jmd.2018007

[4]

Romar dela Cruz, Michael Kiermaier, Sascha Kurz, Alfred Wassermann. On the minimum number of minimal codewords. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020130

[5]

Timothy C. Reluga, Jan Medlock, Alison Galvani. The discounted reproductive number for epidemiology. Mathematical Biosciences & Engineering, 2009, 6 (2) : 377-393. doi: 10.3934/mbe.2009.6.377

[6]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011

[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[8]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[9]

Wilfrid Gangbo, Andrzej Świech. Optimal transport and large number of particles. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1397-1441. doi: 10.3934/dcds.2014.34.1397

[10]

Sujay Jayakar, Robert S. Strichartz. Average number of lattice points in a disk. Communications on Pure & Applied Analysis, 2016, 15 (1) : 1-8. doi: 10.3934/cpaa.2016.15.1

[11]

G.F. Webb. The prime number periodical cicada problem. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 387-399. doi: 10.3934/dcdsb.2001.1.387

[12]

Jared T. Collins. Constructing attracting cycles for Halley and Schröder maps of polynomials. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5455-5465. doi: 10.3934/dcds.2017237

[13]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[14]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437

[15]

Tero Laihonen. Information retrieval and the average number of input clues. Advances in Mathematics of Communications, 2017, 11 (1) : 203-223. doi: 10.3934/amc.2017013

[16]

Françoise Pène. Asymptotic of the number of obstacles visited by the planar Lorentz process. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 567-587. doi: 10.3934/dcds.2009.24.567

[17]

Francesco Fassò, Andrea Giacobbe, Nicola Sansonetto. On the number of weakly Noetherian constants of motion of nonholonomic systems. Journal of Geometric Mechanics, 2009, 1 (4) : 389-416. doi: 10.3934/jgm.2009.1.389

[18]

Qixuan Wang, Hans G. Othmer. The performance of discrete models of low reynolds number swimmers. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1303-1320. doi: 10.3934/mbe.2015.12.1303

[19]

Donatella Donatelli, Bernard Ducomet, Šárka Nečasová. Low Mach number limit for a model of accretion disk. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3239-3268. doi: 10.3934/dcds.2018141

[20]

Radu Saghin. On the number of ergodic minimizing measures for Lagrangian flows. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 501-507. doi: 10.3934/dcds.2007.17.501

2020 Impact Factor: 1.833

Article outline

Figures and Tables

[Back to Top]