Unimodular polynomial matrices over finite fields

Arora, Akansha; Ram, Samrith; Venkateswarlu, Ayineedi

doi:10.1007/s10801-020-00963-2

Cited by 8 publications

(9 citation statements)

References 24 publications

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“…On the other hand 124|35 has interlacing number zero, but is not noncrossing. The arcs (2,4) and (3,5) Table 1 shows the arcs and the number of interlacings for the set partitions in Example 2.7. The first, second, and third arcs are shown in different colours.…”

Section: Definition 33 (Interlacing) Let S Be Any Finite Subset Of P Letmentioning

confidence: 99%

“…Therefore, q 9 c q (T 1 ) + q 7 c q (T 2 ) = q 2 (1 + q)q 7 c q ( T ) + (1 + q)q 7 c q ( T ) = [4] q q 7 c q ( T ).…”

Section: 4mentioning

confidence: 99%

“…Let n = 4 and C = (1, 3). The four multilinear tableaux with support contained in [4] with first column (1,3) and their contributions to q β(C, [4]−C) = q 3 are as follows:…”

Section: 2mentioning

confidence: 99%

“…3.4] gave an explicit formula for the number of ∆-splitting subspaces of dimension m by proving a conjecture of Ghorpade and Ram [12]. Another proof of the result of Chen and Tseng appears in [4] while the case of regular nilpotent ∆ was resolved in [2]. In the case where ∆ has n distinct eigenvalues in F q , Eq.…”

Section: Introductionmentioning

confidence: 99%

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Set partitions, tableaux, and subspace profiles under regular split semisimple matrices

Prasad¹,

Ram²

2021

Preprint

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We introduce a family of univariate polynomials indexed by integer partitions. At prime powers they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. At 1, they count set partitions with specified block sizes. At 0, they count standard tableaux of specified shape. At −1 they count standard shifted tableaux of a specified shape. These polynomials are generated by a new statistic on set partitions (called the interlacing number) as well as a polynomial statistic on standard tableaux. They allow us to express q-Stirling numbers of the second kind as sums over standard tableaux and as sums over set partitions. In a special case these polynomials coincide with those defined by Touchard in his study of crossings of chord diagrams.

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Section: Definition 33 (Interlacing) Let S Be Any Finite Subset Of P Letmentioning

confidence: 99%

“…Therefore, q 9 c q (T 1 ) + q 7 c q (T 2 ) = q 2 (1 + q)q 7 c q ( T ) + (1 + q)q 7 c q ( T ) = [4] q q 7 c q ( T ).…”

Section: 4mentioning

confidence: 99%

“…Let n = 4 and C = (1, 3). The four multilinear tableaux with support contained in [4] with first column (1,3) and their contributions to q β(C, [4]−C) = q 3 are as follows:…”

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Set partitions, tableaux, and subspace profiles under regular split semisimple matrices

Prasad¹,

Ram²

2021

Preprint

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“…Another proof of the result of Chen and Tseng and some connections with unimodular matrices may be found in [3]. Given a linear operator T on V , let σ(m, d; T ) denote the number of m-dimensional T -splitting subspaces.…”

Section: Introductionmentioning

confidence: 99%

Splitting Subspaces of Linear Operators over Finite Fields

Aggarwal¹,

Ram²

2020

Preprint

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Let V be a vector space of dimension N over the finite field F q and T be a linear operator on V . Given an integer m that divides N , an m-dimensional subspace, d; T ) denote the number of m-dimensional T -splitting subspaces. Determining σ(m, d; T ) for an arbitrary operator T is an open problem. We prove that σ(m, d; T ) depends only on the similarity class type of T and give an explicit formula in the special case where T is cyclic and nilpotent. We also show that σ(m, d; T ) is a polynomial in q. Contents 1. Introduction 1 2. Counting Flags of Subspaces 3 3. Similarity Class Type and Splitting Subspaces 6 4. Cyclic Nilpotent Operators 8 5. Polynomiality of σ(m, d; T ) 12 6. Acknowledgements 12 References 12

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Splitting subspaces and a finite field interpretation of the Touchard–Riordan Formula

Prasad

Ram

2023

European Journal of Combinatorics

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Unimodular polynomial matrices over finite fields

Cited by 8 publications

References 24 publications

Set partitions, tableaux, and subspace profiles under regular split semisimple matrices

Set partitions, tableaux, and subspace profiles under regular split semisimple matrices

Splitting Subspaces of Linear Operators over Finite Fields

Splitting subspaces and a finite field interpretation of the Touchard–Riordan Formula

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