Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity

Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity

We give a purely combinatorial proof of the Glaisher-Crofton identity which is derived from the analysis of discrete structures generated by the iterated action of the second derivative. The argument illustrates the utility of symbolic and generating function methodology of modern enumerative combin...

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Main Authors: Pawel Blasiak, Gérard H. E. Duchamp, Karol A. Penson
Format: Article
Language:English
Published: Hindawi Limited 2018-01-01
Series:Advances in Mathematical Physics
Online Access:http://dx.doi.org/10.1155/2018/9575626
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spelling doaj-7c3ee449d9f84c59b7618b2bf9e91ebd2021-07-02T14:46:07ZengHindawi LimitedAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/95756269575626Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton IdentityPawel Blasiak0Gérard H. E. Duchamp1Karol A. Penson2Institute of Nuclear Physics Polish Academy of Sciences, 31342 Kraków, PolandUniversité Paris 13, Sorbonne Paris Cité, LIPN, CNRS UMR 7030, 93430 Villetaneuse, FranceUniversité Pierre et Marie Curie (Paris 06), Sorbonne Universités, LPTMC, CNRS UMR 7600, 75252 Paris Cedex 05, FranceWe give a purely combinatorial proof of the Glaisher-Crofton identity which is derived from the analysis of discrete structures generated by the iterated action of the second derivative. The argument illustrates the utility of symbolic and generating function methodology of modern enumerative combinatorics. The paper is meant for nonspecialists as a gentle introduction to the field of graphical calculus and its applications in computational problems.http://dx.doi.org/10.1155/2018/9575626
collection DOAJ
language English
format Article
sources DOAJ
author Pawel Blasiak
Gérard H. E. Duchamp
Karol A. Penson
spellingShingle Pawel Blasiak
Gérard H. E. Duchamp
Karol A. Penson
Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity
Advances in Mathematical Physics
author_facet Pawel Blasiak
Gérard H. E. Duchamp
Karol A. Penson
author_sort Pawel Blasiak
title Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity
title_short Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity
title_full Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity
title_fullStr Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity
title_full_unstemmed Combinatorics of Second Derivative: Graphical Proof of Glaisher-Crofton Identity
title_sort combinatorics of second derivative: graphical proof of glaisher-crofton identity
publisher Hindawi Limited
series Advances in Mathematical Physics
issn 1687-9120
1687-9139
publishDate 2018-01-01
description We give a purely combinatorial proof of the Glaisher-Crofton identity which is derived from the analysis of discrete structures generated by the iterated action of the second derivative. The argument illustrates the utility of symbolic and generating function methodology of modern enumerative combinatorics. The paper is meant for nonspecialists as a gentle introduction to the field of graphical calculus and its applications in computational problems.
url http://dx.doi.org/10.1155/2018/9575626
work_keys_str_mv AT pawelblasiak combinatoricsofsecondderivativegraphicalproofofglaishercroftonidentity
AT gerardheduchamp combinatoricsofsecondderivativegraphicalproofofglaishercroftonidentity
AT karolapenson combinatoricsofsecondderivativegraphicalproofofglaishercroftonidentity
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