Let S = (s1, s2, . . .) be any sequence of nonnegative integers and let Sk = ∑k i=1 si We then define the falling (rising) factorials relative to S by setting (x)↓k,S= (x−S1)(x−S2) · · · (x−Sk) and (x)↑k,S= (x+S1)(x+S2) · · · (x+Sk) if k ≥ 1 with (x)↓0,S= (x)↑0,S= 1. It follows that {(x)↓k,S}k≥0 and {(x)↑k,S}k≥0 are bases for the polynomial ring Q[x]. We use a rook theory model due to Miceli an...

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With regard to antibiotic resistance studies in various model animals in the urban environment, the presented study focused on the rook, many behavioural and ecological aspects of which are important from an epidemiological point of view. A total of 130 Escherichia coli strains isolated from rook faeces during a two-year period (2011-2012) were investigated for antibiotic resistance and virulen...

Motivated by certain conjectures regarding immanants of Jacobi-Trudi matrices, Stanley has recently defined and studied a symmetric function generalization XG of the chromatic polynomial of a graph G. Independently, Chung and Graham have defined and studied a directed graph invariant called the cover polynomial. The cover polynomial is closely related to the chromatic polynomial and to the rook...

For matrices with a well defined numerical rank in the sense that there is a large gap in the singular value spectrum we compare three rank revealing QR algorithms and four rank revealing LU algorithms with the singular value decomposition. The fastest algorithms are those that construct LU factorizations using rook pivoting. For matrices with a sufficiently large gap in the singular values all...