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Zapiski Nauchnykh Seminarov POMI, 2015, Volume 436, Pages 5–33 (Mi znsl6157)  
Calculation of Pfaffians by a chip removal

V. E. Aksenova, K. P. Kokhasbc

a St. Petersburg National Research University of Information Technologies, Mechanics and Optics, St. Petersburg, Russia
b St. Petersburg National Research University of Information Technologies, Mechanics and Optics, St. Petersburg, Russia
c St. Petersburg State University, St. Petersburg, Russia
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Abstract: We define an operation of chip removal that generalizes the Urban Renewal trick of Kuperberg and Propp. This operation replaces a subgraph $H$ of a graph $G$ with a small collection of weighted edges so that the equalty $\mathrm{Pf}(G)=\mathrm{Pf}(H)\mathrm{Pf}(G')$ holds (here $G'$ is the graph obtained after the replacement). We explain how to calculate the weights of the new edges in terms of the Pfaffians of the chip. We give several applications of this construction. One of these applications is to “Arnold's snakes”, which are graphs with the number of perfect matchings equal to Euler–Bernoulli numbers.
Key words and phrases: Pfaffian, matching number, graphical condensation.
Received: 12.09.2015
English version:
Journal of Mathematical Sciences (New York), 2016, Volume 215, Issue 6, Pages 631–648
DOI: https://doi.org/10.1007/s10958-016-2870-6
Bibliographic databases:
Document Type: Article
UDC: 519.148+519.177.3
Language: Russian
Citation: V. E. Aksenov, K. P. Kokhas, “Calculation of Pfaffians by a chip removal”, Representation theory, dynamical systems, combinatorial methods. Part XXV, Zap. Nauchn. Sem. POMI, 436, POMI, St. Petersburg, 2015, 5–33; J. Math. Sci. (N. Y.), 215:6 (2016), 631–648
Citation in format AMSBIB
\Bibitem{AksKok15}
\by V.~E.~Aksenov, K.~P.~Kokhas
\paper Calculation of Pfaffians by a~chip removal
\inbook Representation theory, dynamical systems, combinatorial methods. Part~XXV
\serial Zap. Nauchn. Sem. POMI
\yr 2015
\vol 436
\pages 5--33
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl6157}
\mathscinet{https://mathscinet.ams.org/mathscinet-getitem?mr=3498183}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2016
\vol 215
\issue 6
\pages 631--648
\crossref{https://doi.org/10.1007/s10958-016-2870-6}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84966667749}
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