Corpus ID: 212657361

A symmetric polynomial determinant identity generalising a binomial identity of Gessel and Viennot and a symmetric function identity of Aitken

@article{McDowell2020ASP,
  title={A symmetric polynomial determinant identity generalising a binomial identity of Gessel and Viennot and a symmetric function identity of Aitken},
  author={Eoghan McDowell},
  journal={arXiv: Combinatorics},
  year={2020}
}
This paper proves a symmetric polynomial determinant identity which generalises both the binomial determinant duality theorem due to Gessel and Viennot and the symmetric function duality theorem due to Aitken. As corollaries we obtain the lifts of the binomial determinant duality theorem to \(q\)-binomial coefficients and to symmetric polynomials. Our method is a path counting argument on a novel lattice generalising that used by Gessel and Viennot. 

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References

SHOWING 1-10 OF 14 REFERENCES
Note on Dual Symmetric Functions
§ 1. In an earlier paper, which this note is intended to supplement and in some respects improve, the writer gave a general theorem of duality relating to isobaric determinants with elements Cr andExpand
Symmetric functions and Hall polynomials
I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functionsExpand
Algebraic/combinatorial proofs of Cayley-type identities for derivatives of determinants and pfaffians
TLDR
This paper presents straightforward algebraic/combinatorial proofs of a variety of Cayley-type identities, both old and new, that employ Grassmann algebra (=exterior algebra) and Grassmann-Berezin integration. Expand
Binomial Determinants, Paths, and Hook Length Formulae
Abstract We give a combinatorial interpretation for any minor (or binomial determinant) of the matrix of binomial coefficients. This interpretation involves configurations of nonintersecting paths,Expand
On Stirling numbers
We exhibit an approach for the recursive formula obtained by Cereceda for the sums of powers of integers. Besides, we show that the definitions of Janjic for the Stirling numbers are consequences ofExpand
Counting on Determinants
TLDR
Imagine four determined ants who simultaneously walk along the edges of the picnic table graph of Figure 1.1 with the goal of reaching four different morsels. Expand
American Mathematical Monthly
s for articles or notes should entice the prospective reader into exploring the subject of the paper and should make it clear to the reader why this paper is interesting and important. The abstractExpand
Paths
  • The Science of Play
  • 2014
Why should wait for some days to get or receive the paths book that you order? Why should you take it if you can get the faster one? You can find the same book that you order right here. This is itExpand
Oxford classic texts in the physical sciences
  • 1998
...
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