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doi: 10.3934/era.2021018

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 Published  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

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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

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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

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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

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

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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
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