Determinant formulas of some Toeplitz–Hessenberg matrices with Catalan entries

Goy, Taras; Shattuck, Mark

doi:10.1007/s12044-019-0513-9

Cited by 8 publications

(14 citation statements)

References 5 publications

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“…Example 1. When n = 4, formula (7) yields (8) and ( 9) respectively that P 5 3 − 4P 3 3 P 4 + 3P 2 3 P 5 + 3P 3 P 2 4 − 2P 3 P 6 − 2P 4 P 5 + P 7 = 2F 0 , P 6 3 − 5P 4 3 P 5 + 4P 3 3 P 7 + 6P 2 3 P 2 5 − 3P 2 3 P 9 − 3P 3 P 5 P 7 + 2P 3 P 11 − P 3 5 + 2P 5 P 9 + P 2 7 − P 13 = −F 2 .…”

Section: Applications Of Trudi's Formulamentioning

confidence: 99%

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A note on Pell-Padovan numbers and their connection with Fibonacci numbers

Goy

Sharyn

2020

Carpathian Math. Publ.

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In this paper, we find new relations involving the Pell-Padovan sequence which arise as determinants of certain families of Toeplitz-Hessenberg matrices. These determinant formulas may be rewritten as identities involving sums of products of Pell-Padovan numbers and multinomial coefficients. In particular, we establish four connection formulas between the Pell-Padovan and the Fibonacci sequences via Toeplitz-Hessenberg determinants.

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Section: Applications Of Trudi's Formulamentioning

confidence: 99%

“…Using (2), now we obtain from Corollary 1 the following multinomial Fibonacci identities (many identities with Fibonacci numbers and their generalizations are given in [5,[8][9][10][11][12]). Corollary 3.…”

Section: Applications Of Trudi's Formulamentioning

confidence: 99%

A note on Pell-Padovan numbers and their connection with Fibonacci numbers

Goy

Sharyn

2020

Carpathian Math. Publ.

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Section: Introductionmentioning

confidence: 99%

“…which was later rediscovered by the authors in [8] (formula (2.21) with a = −1). In [8], the authors found determinants of several families of Toeplitz-Hessenberg matrices having various subsequences of the Catalan sequence for the nonzero entries. These determinant formulas may also be rewritten equivalently as identities involving sums of products of Catalan numbers and multinomial coefficients.…”

Section: Introductionmentioning

confidence: 99%

Hessenberg-Toeplitz Matrix Determinants with Schroder and Fine Number Entries

Goy¹,

Shattuck²

2023

Preprint

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In this paper, we find determinant formulas of several Hessenberg-Toeplitz matrices whose nonzero entries are derived from the small and large Schröder and Fine number sequences. Algebraic proofs of these results can be given which make use of Trudi's formula and the generating function of the associated sequence of determinants. We also provide direct arguments of our results that utilize various counting techniques, among them sign-changing involutions, on combinatorial structures related to classes of lattice paths enumerated by the Schröder and Fine numbers. As a consequence of our results, we obtain some new formulas for the Schröder and Catalan numbers as well as for some additional sequences from the OEIS in terms of determinants of certain Hessenberg-Toeplitz matrices.

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“…Excellent sources on these numbers are the books by Koshy [13], Roman [18] and Stanley [21]. Some examples of recent work involving Catalan numbers and their generalizations include [5,7,8,10,11,15,16,17,24].…”

Section: Introductionmentioning

confidence: 99%