Tableau complexes

Knutson, Allen; Miller, Ezra; Yong, Alexander

doi:10.1007/s11856-008-0014-5

Cited by 18 publications

(28 citation statements)

References 12 publications

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“…The proof in this section is completely combinatorial, while the original proofs in Knutson-Miller-Yong [15] and Hudson-Matsumura [12] are geometric. On the other hand, Knutson-Miller-Yong also gives a combinatorial proof of their tableau formula in [14] by showing that it satisfies Lascoux's transition formula for Grothendieck polynomials while ours is an alternative combinatorial proof using the compatibility of the tableau formula with the divided difference operators.…”

Section: Grothendieck Polynomialsmentioning

confidence: 87%

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Flagged Grothendieck polynomials

Matsumura

2018

J Algebr Comb

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We show that the flagged Grothendieck polynomials defined as generating functions of flagged set-valued tableaux of Knutson et al. (J Reine Angew Math 630:1-31, 2009) can be expressed by a Jacobi-Trudi-type determinant formula generalizing the work of Hudson-Matsumura (Eur J Comb 70:190-201 2018). We also introduce the flagged skew Grothendieck polynomials in these two expressions and show that they coincide.

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Section: Grothendieck Polynomialsmentioning

confidence: 87%

“…Motivated by these results, we study the generating functions of flagged set-valued tableaux in general, beyond the ones given by vexillary permutations. We follow the works [14,15] of Knutson-Miller-Yong. For a given partition λ = (λ 1 ≥ • • • ≥ λ r > 0), a flagging f = ( f 1 , .…”

Section: Introductionmentioning

confidence: 99%

Flagged Grothendieck polynomials

Matsumura

2018

J Algebr Comb

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“…be sets of infinitely many indeterminants. In [20] (see also [19]), Knutson-Miller-Yong described the double Grothendick polynomials of Lascoux-Schützenberger [24] associated to a vexillary permutation w as a generating function of flagged set-valued tableaux. Namely we have…”

Section: Vexillary Double Grothendieck Polynomialsmentioning

confidence: 99%

Vexillary degeneracy loci classes in K-theory and algebraic cobordism

Hudson

Matsumura

2018

European Journal of Combinatorics

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“…An assignment tableau s λ represents a task assignment, where each task in a cell (i, j) of t λ is assigned to each agent in a cell (i, j) of a λ bijectively. Therefore, we also denote s λ as the set of all (a → b) [25], where a is a task in a cell (i, j) of t λ and b is an agent in a cell (i, j) of a λ . Definition 5.3.…”

Section: Assignment Tableaux and Tabloids For N-task-n-agent Assignmentsmentioning

confidence: 99%

Representations of task assignments in distributed systems using Young tableaux and symmetric groups

Kim¹

2015

International Journal of Parallel, Emergent and Distributed Sys

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This paper presents a novel approach to representing task assignments for partitioned agents (respectively, tasks) in distributed systems. A partition of agents (respectively, tasks) is represented by a Young tableau, which is one of the main tools in studying symmetric groups and combinatorics. In this paper we propose a task, agent, and assignment tableau in order to represent a task assignment for partitioned agents (respectively, tasks) in a distributed system. This paper is concerned with representations of task assignments rather than finding approximate or near optimal solutions for task assignments. A Young tableau approach allows us to raise the expressiveness of partitioned agents (respectively, tasks) and their task assignments.1 In this paper we use agent and processor interchangeably if parallel agents in a distributed system are considered as simple computing entities [44]. A task graph T = (V, E) is a directed acyclic graph, where each node in V = {v 1 , v 2 , . . . , v n } denotes a task, and each edge (v i , v j ) ∈ E ⊂ V × V denotes a precedence relationship between tasks, i.e., v j cannot begin before v i completes. The positive weight associated with each task v ∈ V represents a computation requirement. The nonnegative weight associated with each edge (v i , v j ) ∈ E represents a communication requirement.A fully-connected heterogeneous system A is a set A = {a 1 , a 2 , . . . , a m } of m heterogeneous agents whose network topology is fully-connected. A heterogeneous system A is called consistent if agent a i ∈ A executes a task n-times faster than agent a j ∈ A, then it executes all other tasks n-times faster than agent a j . A heterogeneous system A is called communication homogeneous if each communication link in A is identical.Let T = (V, E) be a task graph and A = {a 1 , a 2 , . . . , a m } be a fully-connected heterogeneous system. Assume that a startup cost of initiating a task on an agent is negligible. In a consistent system, the computation cost of taskis the computation requirement of task v i , and e(a j ) is the execution rate of agent a j . Meanwhile, in an inconsistent system, the computation cost of task v i on a j is given by ω(v i , a j ) = w ij , where w ij is the (i, j) th entry in a |V | × |A| cost matrix W . Note that an inconsistent system model is a generalization of a consistent system model. Now, the communication cost model of A is defined as follows. Let d(v i , v j ) be the amount of data to be transferred from task v i to task v j for each (v i , v j ) ∈ E; let t(a s , a t ) be the data transfer rate between the communication link between agent a s and a t in A. Let M be a task assignment between T and A such that M (v i ) = a s and M (v j ) = a t for v i , v j ∈ T and a s , a t ∈ A. Assume that both local communication and communication startup cost are negligible. If the communication of A has the linear cost model [2], the communication cost between task v i on agent a s and task v j on a t is given by c(M (v i ), M (v j )) = d(v i , v j )/t(a s , a t ) if a ...

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Tableau complexes

Cited by 18 publications

References 12 publications

Flagged Grothendieck polynomials

Flagged Grothendieck polynomials

Vexillary degeneracy loci classes in K-theory and algebraic cobordism

Representations of task assignments in distributed systems using Young tableaux and symmetric groups

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