# American Institute of Mathematical Sciences

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 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:
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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. Adin, E. 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. Chung, R. L. Graham, V. 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. Claesson, S. 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. Dokos, T. Dwyer, B. P. Johnson, B. 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. Fu, G. -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. Fu, Z. 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
First three levels of the generating tree for $\cup_{n\geq1}{\mathfrak{S}}_n(2431, 4231)$
The block decomposition of separable permutations
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
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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