The umbral transfer-matrix method. I: Foundations. In this paper the author lays the foundation for the umbral transfer-matrix method based on G. C. Rota’s realization of an umbra, merely a linear functional on a vector space of formal power series. It appears to be the first in a series of papers to be written by the author aiming to show how Rota’s concept blended with the transfer-matrix method could be gainfully employed to compute generating functions for many difficult problems dealing with counting combinatorial objects.

References in zbMATH (referenced in 18 articles )

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  1. Drmota, Michael; Noy, Marc; Yu, Guan-Ru: Universal singular exponents in catalytic variable equations (2022)
  2. Gao, Zhicheng; MacFie, Andrew: Locally restricted compositions over a finite group (2021)
  3. Mansour, Toufik; Rastegar, Reza: Convex polyominoes revisited: enumeration of outer site perimeter, interior vertices, and boundary vertices of certain degrees (2020)
  4. Beaton, Nicholas R.; Bouvel, Mathilde; Guerrini, Veronica; Rinaldi, Simone: Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers (2019)
  5. Bernardi, Olivier; Bousquet-Mélou, Mireille: Counting coloured planar maps: differential equations (2017)
  6. Chen, Joanna N.; Li, Shouxiao: A new bijective proof of Babson and Steingrímsson’s conjecture (2017)
  7. Baxter, Andrew; Nakamura, Brian; Zeilberger, Doron: Automatic generation of theorems and proofs on enumerating consecutive-Wilf classes (2013)
  8. Bousquet-Mélou, Mireille: Counting planar maps, coloured or uncoloured (2011)
  9. Bousquet-Mélou, Mireille; Claesson, Anders; Dukes, Mark; Kitaev, Sergey: (2+2)-free posets, ascent sequences and pattern avoiding permutations (2010)
  10. Bernardi, Olivier: On triangulations with high vertex degree (2008)
  11. Ekhad, Shalosh B.; Zeilberger, Doron: Using Rota’s Umbral calculus to enumerate Stanley’s (P)-partitions (2008)
  12. Petrullo, P.; Senato, D.: An instance of umbral methods in representation theory: the parking function module (2008)
  13. Bousquet-Mélou, Mireille; Jehanne, Arnaud: Polynomial equations with one catalytic variable, algebraic series and map enumeration (2006)
  14. Bousquet-Mélou, Mireille: Algebraic generating functions in enumerative combinatorics and context-free languages (2005)
  15. Bousquet-Mélou, M.; Rechnitzer, A.: The site-perimeter of bargraphs (2003)
  16. Zeilberger, Doron: The umbral transfer-matrix method. III: Counting animals (2001)
  17. Zeilberger, Doron: The umbral transfer-matrix method. IV: Counting self-avoiding polygons and walks (2001)
  18. Zeilberger, Doron: The umbral transfer-matrix method. I: Foundations (2000)