On Goulden-Jackson’s Determinantal Formula for the Immanant

We will determine the number of powers ofαthat appear with nonzero coefficient in anα-power linear differential resolvent of smallest possible order of a univariate polynomialP(t)whose coefficients lie in an ordinary differential field and whose distinct roots are differentially independent over constants. We will then give an upper bound on the weight of anα-resolvent of smallest possible weight. We will then compute the indicial equation, apparent singularities, and Wronskian of the Cockleα-resolvent of a trinomial and finish with a related determinantal formula.

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Explicit determinantal formula for a class of banded matrices

Open Mathematics ◽

10.1515/math-2020-0100 ◽

2020 ◽

Vol 18 (1) ◽

pp. 1227-1229

Author(s):

Yerlan Amanbek ◽

Zhibin Du ◽

Yogi Erlangga ◽

Carlos M. da Fonseca ◽

Bakytzhan Kurmanbek ◽

...

Keyword(s):

Short Note ◽

Determinantal Formula ◽

Banded Matrices

Abstract In this short note, we provide a brief proof for a recent determinantal formula involving a particular family of banded matrices.

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Determinantal formula for generalized riffle shuffle

Discrete Mathematics ◽

10.1016/j.disc.2021.112599 ◽

2021 ◽

Vol 344 (12) ◽

pp. 112599

Author(s):

Fumihiko Nakano ◽

Taizo Sadahiro

Keyword(s):

Determinantal Formula

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A quantum-trace determinantal formula for matrix commutators, and applications

Linear Algebra and its Applications ◽

10.1016/j.laa.2011.07.010 ◽

2012 ◽

Vol 436 (7) ◽

pp. 2380-2397

Author(s):

Dinesh Khurana ◽

T.Y. Lam ◽

Noam Shomron

Keyword(s):

Determinantal Formula

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Determinantal Expression and Recursion for Jack Polynomials

The Electronic Journal of Combinatorics ◽

10.37236/1539 ◽

1999 ◽

Vol 7 (1) ◽

Cited By ~ 6

Author(s):

Luc Lapointe ◽

A. Lascoux ◽

J. Morse

Keyword(s):

Schur Functions ◽

Jack Polynomials ◽

Efficient Computation ◽

Monomial Basis ◽

Determinantal Formula ◽

Kostka Numbers ◽

Determinantal Expression ◽

Monomial Functions

We describe matrices whose determinants are the Jack polynomials expanded in terms of the monomial basis. The top row of such a matrix is a list of monomial functions, the entries of the sub-diagonal are of the form $-(r\alpha+s)$, with $r$ and $s \in {\bf N^+}$, the entries above the sub-diagonal are non-negative integers, and below all entries are 0. The quasi-triangular nature of these matrices gives a recursion for the Jack polynomials allowing for efficient computation. A specialization of these results yields a determinantal formula for the Schur functions and a recursion for the Kostka numbers.

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