A folding phenomenon on partitions

Hsiang-Chun Hsu

Department of Mathematics

Tamkang University

hchsu0222@gmail.com

    In this talk we will introduce the signed `q`-counting over partitions whose Ferrers diagrams fit inside a given partition, where the sign is the parity of the size and the enumerator statistic is the length. We will introduce several `q`-identities, exhibiting a certain pattern which we called the folding phenomenon.

Keyword: partition, Ferrers diagram, `q`-analogue, folding phenomenon

References

[1] R.M. Adin, Y. Roichman, Equidistribution and sign-balance on 321-avoiding permutations, Sémin. Loth. Combin. 51 (2004) B51d.
[2] W.Y.C. Chen, L.W. Shapiro, L.L.M. Yang, Parity reversing involution on plane trees and 2-Motzkin paths, European J. Combin. 27 (2006) 283–289.
[3] J. Désarménien, D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992) 49–58.
[4] S.-P. Eu, S.-C. Liu, Y.-N. Yeh, Odd or even on plane trees, Discrete Math. 281 (2004) 189–196.
[5] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Sign-balance identities of Adin-Roichman type on 321-avoiding alternating permutations, Discrete Math. 312 (2012) 2228–2237.
[6] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Baxter Permutations, Maj-balances, and Positive Braids, Electronic J. Combin. 19 Issue 3 (2012) P26.
[7] S.-P. Eu, T.-S. Fu, Y.-J. Pan, C.-T. Ting, Two refined major-balance identities on 321-avoiding involutions European J. Combin. 49 (2015) 250–264.
[8] S.-P. Eu, T.-S. Fu, Y.-J. Pan, P.-L. Yan, More on double Simsun permutations, non-published manuscript.
[9] T. Mansour, Equidistribution and sign-balance on 132-avoiding permutations, Sémin. Loth. Combin. 51 (2004) B51e.
[10] A. Reifegerste, Refined sign-balance on 321-avoiding permutations, European J. Combin. 26 (2005) 1009–1018.
[11] A. Robertson, D. Saracino, D. Zeilberger, Refined restricted permutations, Ann. Comb. 6 (2002), 427–444.
[12] M. Shattuck, Parity theorems for statistics on permutations and Catalan words, Integers: Electronic J. Combin. Number Theory 5 (2005) #A07.
[13] R. Simion, F.W. Schmidt, Restricted permutations, European J. Combin. 6 (1985) 383–406.
[14] R. Stanley. Some remarks on sign-balanced and maj-balanced posets. Adv. Appl. Math. 34(4) (2005) 880–902.
[15] C.-T. Ting, Folding phenomena of some classes of permutations, Thesis, 2017.
[16] M. Wachs, An involution for signed Eulerian numbers, Discrete Math. 99 (1992) 59–62.