Antonio Bernini

Biography

Antonio Bernini received his PhD in Computer Engineering and Automation at the University of Florence (Coordinator Prof. Mosca) with a thesis entitled "Some properties of pattern avoiding permutations" (Advisor: Prof. R. Pinzani). He received his Master's degree in Mathematics on September 19th, 2003 at the University of Florence with full marks (cum laude) with a thesis entitled "Enumerations of permutations avoiding three patterns" (Advisor: Prof. Pinzani). He received his university degree in Mathematics in 2001 at the University of Florence with full marks (cum laude) with a thesis titled "A Scientific Approach for Evaluation of Consonance in the Musical Scales" (Advisor: Prof. Talamucci). In 1997 he received his academic degree in "Choral Music and Choir Conducting" at the Conservatory of Music "L. Cherubini" of Florence. In 1996 he received his academic degree in "Composing" at the Conservatory of Music "L.Cherubini" of Florence.

Research Interest

Antonio Bernini works within the group of Discrete Mathematics and theoretical Computer Science under the supervision of Prof. R. Pinzani at the System and Computer Science Department of the University of Florence. One of the topics where some interesting results have been found is pattern avoiding permutations, more specifically generalized pattern avoiding permutations. Generalized patterns have been introduced by Babson and Steingrìmsson [BS] for the study of the mahonian statistics of permutations. Using ECO method [BDPP1] and a particular graphical representation of permutations all the conjectures by Claesson and Mansour [CM] on the enumeration of permutations avoiding three Babson-Steingrīmsson patterns (generalized patterns of length three) have been solved in positive sense. The used approach has provided some simple and nice applications of ECO method. The results are in [BFP2]. Afterwords, also the conjectures on cases of more than three generalized forbidden patterns have been solved [BP]: using some propositions on generalized pattern voiding permutations the most cases of the general problem have been treated, also with the help of the results of [BFP]. An interesting result has been pointed out during this study: considering permutation avoiding the pattern 1-23 we have provided their distribution according to the length and the value of the last element. This result has been generalized to the permutations that avoid any generalized pattern of length three and to some cases of permutations avoiding two generalized patterns. The results of this investigation are in [BBF]. considering permutation avoiding the pattern 1-23 we have provided their distribution according to the length and the value of the last element. This result has been generalized to the permutations that avoid any generalized pattern of length three and to some cases of permutations avoiding two generalized patterns. The results of this investigation are in [BBF]. considering permutation avoiding the pattern 1-23 we have provided their distribution according to the length and the value of the last element. This result has been generalized to the permutations that avoid any generalized pattern of length three and to some cases of permutations avoiding two generalized patterns. The results of this investigation are in [BBF]. The approach used in [BFP2] (and in [BP]) for the enumeration of generalized pattern avoiding permutations can also be useful for words. More specifically, it is possible to give a general procedure for generating words avoiding generalized patterns and often it is possible to describe it by a succession rule. Ten, in several cases, we are able to provide a closed formula formula enumerating the considered words or the generating function of the class of words according to their length. In [BFP1] some results on the enumeration of words avoiding two generalized patterns of length three are found. In the same research area of ​​permutations, also classical patterns have been considered. In particular, some classes of permutations have been introduced such that their enumerating sequences are limited by the Fibonacci and Catalan sequences. This result can be reached by considering the classes S (123, 213, 312) where the patterns 213 and 312 are seen as specific cases of increasing length patterns so that the final class to be considered is S (123) (enumerated by Catalano numbers). These results are in [BBP] and continue with a research line already existing [BDPP2, BDPP3] where the idea of ​​discrete continuity between number sequences appeared. A different topic is the study of order properties of classes of combinatorial objects (permutations, paths, partitions, words). A first interesting result [BBFP] in this direction connects Dyck paths, 312 avoiding permutations and noncrossing partitions. Transferring the natural order relation of Dyck paths on noncrossing partitions, we define a distributive structure on them and we prove that the 312 avoiding permutations (in bijection with Dyck paths) form a distributive subposet of the general poset of all the permutations with the strong Bruhat order. A similar study has been carried out from the Motzkin and Schröder paths [BF]: it is possible to define a distribution structure poset on a subset of noncrossing partitions and on certain classes of pattern avoiding permutations (k - (k - 1) (k - 2)... A partial order relation is defined on permutations: the permutation is less than a second permutation if the first one is contained in the second one as a pattern. An interesting question is the study of the Möbius function of the resulting poset. Partial results have been achieved [BJJS, SV, ST]. In general, the poset is quite complicated to analyze. So, we have considered the consecutive pattern poset, where the partial order relation requires that a permutation is less than a second permutation if the first one is contained in the second one as consecutive pattern. This restriction leads to a more simple poset (each permutation covers at most two other permutations) and a procedure for computing the Möbius function of any interval in this poset has been provided. The results are in [BFS]. Other results have been obtained in the field of exhaustive generation of combinatorial objects, showing that all the Dyck paths of a fixed length can be generated by a CAT algorithm [BFG]. The algorithm can be generalized to Grand Dyck and Motzkin paths. In the same research area, we have defined a procedure to list the objects enumerated by the catalan numbers, starting from a particular succession rule. This procedure generates a list where two consecutive codes differ only for a digit, so that a Gray code is obtained. This methodology can be generalized to a more general group of succession rules having a particular property that we have defined stability property. He has been a member of the national research project "Automation and formal languages: mathematical and application aspects" (PRIN 2005). He has been a member of the national research project "Mathematical Aspects and Emerging Applications of Automata and Formal Languages" (PRIN 2007). He has been invited by Prof. Manuel Castellet, director of the Center for Recruiting Mtematica for two months (May and June 2007) in the "Research Program on Enumerative Combinatorics and Random Structures" coordinated by Prof. Marc Noy and Prof. Dominic Welsh. He has been a member of the organizing committee of the international conference "Permutation Patterns 2009" (Florence, July 13-17, 2009) and "Lattice Path Combinatorics and Applications" (Siena, July 4-7, 2010). He has been referee for the journals "Discrete Mathematics", "Ars Combinatoria", and "Pure Mathematics and Applications". He is a reviewer of "Mathematical Reviews".