Rook Numbers Research Articles

Analysis of the Rook (<em>Corvus frugilegus</em>) wintering population in the Friuli Venezia Giulia region (Italy) using maximum entropy models

This research analyzed the wintering population of rooks (Corvus frugilegus) in the Friuli Venezia Giulia region (Italy) from 2010 to 2020. The goal was to determine the population size, trend, and distribution over the decade by assessing census data and evaluating the study area's environmental suitability. It was important to identify the relationship between the region’s environmental characteristics and the presence of the species by developing territory suitability models, processed using the maximum entropy algorithms and MaxEnt (maximum entropy) software. It was possible to determine the combinations of the factors potentially influencing habitat attendance, to develop environmental suitability maps, and to identify the area with the greater probability of the presence of the species, as well as those areas with potential suitability. Data for the presence of wintering rooks were derived from annual censuses, carried out by direct observation along pre-established transects in the Friuli Venezia Giulia region. Based on the census data, that total number of rooks in Friuli Venezia Giulia across the 10-year period had a maximum peak recorded in 2015 (1543) and a minimum in 2019 (853). The distribution of the species was concentrated in the central-southern areas of the region, with greater turnout in the areas characterized by agricultural land use, while the mountainous areas, located in the southern part of the region, were excluded. The environmental suitability models developed with MaxEnt on a regional scale demonstrated excellent predictive capacity, validating the species’ ecological needs. Although the rook is not an endangered species, its range has been drastically reduced in Italy, at least in the last century, and MaxEnt could be used in the future as a management tool in order to outline conservation actions for the species, especially for its habitats.

On the Maximum Number of Non-attacking Rooks on a High-Dimensional Simplicial Chessboard

The simplicial rook graph \({\mathcal {SR}}(m,n)\) is the graph whose vertices are vectors in \( {\mathbb {N}}^m\) such that for each vector the summation of its coordinates is n and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. Martin and Wagner (Graphs Combin 31:1589–1611, 2015) asked about the independence number of \({\mathcal {SR}}(m,n)\) that is the maximum number of non-attacking rooks which can be placed on a \((m-1)\)-dimensional simplicial chessboard of side length \(n+1\). In this work, we solve this problem and show that \(\alpha ({\mathcal {SR}}(m,n))=\big (1-o(1)\big )\frac{\left( {\begin{array}{c}n+m-1\\ n\end{array}}\right) }{m}\). We also prove that for the domination number of rook graphs we have \(\gamma ({\mathcal {SR}}(m, n))= \varTheta (n^{m-2})\). Moreover we show that these graphs are Hamiltonian. The cyclic simplicial rook graph \({\mathcal {CSR}}(m,n)\) is the graph whose vertices are vectors in \({\mathbb {Z}}^{m}_{n}\) such that for each vector the summation of its coordinates modulo n is 0 and two vertices are adjacent if their corresponding vectors differ in exactly two coordinates. In this work we determine several properties of these graphs such as independence number, chromatic number and automorphism group. Among other results, we also prove that computing the distance between two vertices of a given \({\mathcal {CSR}}(m,n)\) is \( \mathbf {NP}\)-hard in terms of n and m.

A Graph Theory of Rook Placements

Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph on an equivalence class of Ferrers boards by specifying that two boards are connected by an edge if you can obtain one of the boards by moving squares in the other board out of one column and into a single other column. Given such a graph, we characterize which boards will yield connected graphs. We also provide some cases where common graphs will or will not be the graph for some set of rook equivalent Ferrers boards. Finally, we extend this graph definition to the m-level rook placement generalization developed by Briggs and Remmel. This yields a graph on the set of rook equivalent singleton boards, and we characterize which singleton boards give rise to a connected graph.

Regularity and the Gorenstein property of $L$-convex Polyominoes

We study the coordinate ring of an $L$-convex polyomino, determine its regularity in terms of the maximal number of rooks that can be placed in the polyomino. We also characterize the Gorenstein $L$-convex polyominoes and those which are Gorenstein on the punctured spectrum, and compute the Cohen–Macaulay type of any $L$-convex polyomino in terms of the maximal rectangles covering it. Though the main results are of algebraic nature, all proofs are combinatorial.

Art Gallery Problem with Rook and Queen Vision

How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that lfloor {frac{n}{2}} rfloor rooks or lfloor {frac{n}{3}} rfloor queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

Eulerian polynomials via the Weyl algebra action

Through the action of the Weyl algebra on the geometric series, we establish a generalization of the Worpitzky identity and new recursive formulae for a family of polynomials including the classical Eulerian polynomials. We obtain an extension of the Dobiński formula for the sum of rook numbers of a Young diagram by replacing the geometric series with the exponential series. Also, by replacing the derivative operator with the q-derivative operator, we extend these results to the q-analogue setting including the q-hit numbers. Finally, a combinatorial description and a proof of the symmetry of a family of polynomials introduced by one of the authors are provided.

Elliptic rook and file numbers

In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corre- sponding factorization theorems which in the classical case were established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind as well as Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel.

Winter habitat selection of Corvids in an urban ecosystem

Urban areas attract birds during the winter when cities provide a predictable source of food and relatively stable weather conditions. However, many other factors determine the occurrence of birds in cities. This study analysed the relationship between corvids wintering in the city of Poznań, in western Poland and habitat features. Each of 32 research site was investigated three times. The most abundant species was the rook (mean = 17.4 individuals/site); the most frequent was the Eurasian magpie (88% of all sites). To determine variables that influence the abundance of birds, a set of generalised linear models was created, with model averaging being selected for each species. The abundance of hooded crows was positively influenced by surface of parks, water and by the length of tram tracks. The number of rooks was positively linked with the number of walnut trees, garbage cans and the surface of the water and negatively linked with the length of watercourses. Jackdaw was affected positively by the number of garbage cans, the surface of the water and negatively by the cover of trees. The number of magpies was associated with the number of walnut trees, garbage cans and negatively influenced by the surface of trees. None of the investigated habitat variables significantly affected the abundance of European jay. We also found a positive correlation between the abundances of hooded crow, rook, jackdaw and magpie. The results indicate that corvids wintering in urban ecosystems exhibit selectivity toward specific habitat features and that preferences differ among the studied species.

Total non-negativity of some combinatorial matrices

Many combinatorial matrices — such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers — are known to be totally non-negative, meaning that all minors (determinants of square submatrices) are non-negative.The examples noted above can be placed in a common framework: for each one there is a non-decreasing sequence (a1,a2,…), and a sequence (e1,e2,…), such that the (m,k) entry of the matrix is the coefficient of the polynomial (x−a1)⋯(x−ak) in the expansion of (x−e1)⋯(x−em) as a linear combination of the polynomials 1,x−a1,…,(x−a1)⋯(x−am).We consider this general framework. For a non-decreasing sequence (a1,a2,…) we establish necessary and sufficient conditions on the sequence (e1,e2,…) for the corresponding matrix to be totally non-negative. As corollaries we obtain total non-negativity of matrices of rook numbers of Ferrers boards, and of graph Stirling numbers of chordal graphs.

Chained permutations and alternating sign matrices—Inspired by three-person chess

We define and enumerate two new two-parameter permutation families, namely, placements of a maximum number of non-attacking rooks on k chained-together n×n chessboards, in either a circular or linear configuration. The linear case with k=1 corresponds to standard permutations of n, and the circular case with n=4 and k=6 corresponds to a three-person chessboard. We give bijections of these rook placements to matrix form, one-line notation, and matchings on certain graphs. Finally, we define chained linear and circular alternating sign matrices, enumerate them for certain values of n and k, and give bijections to analogues of monotone triangles, square ice configurations, and fully-packed loop configurations.

Elliptic Rook and File Numbers

Utilizing elliptic weights, we construct an elliptic analogue of rook numbers for Ferrers boards. Our elliptic rook numbers generalize Garsia and Remmel's $q$-rook numbers by two additional independent parameters $a$ and $b$, and a nome $p$. The elliptic rook numbers are shown to satisfy an elliptic extension of a factorization theorem which in the classical case was established by Goldman, Joichi and White and extended to the $q$-case by Garsia and Remmel. We obtain similar results for elliptic analogues of Garsia and Remmel's $q$-file numbers for skyline boards. We also provide an elliptic extension of the $j$-attacking model introduced by Remmel and Wachs. Various applications of our results include elliptic analogues of (generalized) Stirling numbers of the first and second kind, Lah numbers, Abel numbers, and $r$-restricted versions thereof.

Spatial patterns of habitat distribution of Corvidae (the case of urban-rural gradient)

The spatial distribution and abundance of Corvidae species in Zhytomyr region was studied in terms of the urban-rural gradient. We selected Rook (Corvus frugilegus L.), Eurasian Jackdaw (C. monedula L.), Hooded Crow (C. cornix L.), Common Raven (C. corax L.), Eurasian Magpie (Pica pica L.) and Eurasian Jay (Garrulus glandarius L.) for our observations during 2009–2012. Some 38 survey routes totaling more than 8000 km were surveyed in 21 settlements in Zhytomyr region. Among them 13 routes were in Zhytomyr city. The average density of Rooks was 55.9 birds/km2. We found a strong correlation between Rook density and rural-urban gradient and observed that the numbers of wintering Rooks in cities significantly increased due to the influx from villages. The peak number of Rooks in villages was registered in the breeding and post-breeding seasons while in the cities it was high in winter and during the spring migration. The average density of Eurasian Magpies in the study area was 8.7 birds/km2 and we registered weak correlation with the urban-rural gradient. Their maximum density in urban habitats was in the winter period whereas the highest density in rural habitats was fixed in the summer months. The density of Magpies varied insignificantly within a narrow range during the three years of research, which suggests the species has successfully adjusted to the transformed landscapes. Average density of Hooded Crows in towns was 6.6 birds/km2. The linear relationship between the urban-rural gradient and the density of this species was rather weak. In the breeding period, the birds’ density was considerably higher in urban habitats. We also registered that the average density of Hooded Crows changed insignificantly but gradually increased during the study period. The average density of Eurasian Jackdaws was 9.7 birds/km2 and had high annual dynamics. It also changed significantly during the three years of research. The average density of Eurasian Jays was 2.4 birds/km2 and varied significantly in different settlements, although the fluctuation range was small (0.1–9.3 birds/km2). We registered an expansion in the breeding distribution of Eurasian Jays in recent years i.e. the birds began to nest in old urban neighborhoods and small central parks of large cities. However, the number of birds naturally decreased along the gradient of landscape transformation. The impact of urban-rural gradient on Eurasian Jay density was significant but the correlation was weak and negative. The density of this species was highest in small villages (4.9 birds/km2) and the lowest in medium-sized cities (1.5 birds/km2). The seasonal density of Jays varied significantly and the greatest value was registered in the post-breeding period. In urban areas the birds’ density was much higher in winter compared to the nesting period, and in villages we observed the reverse situation. The number of Eurasian Jays remained almost stable in all the settlements during the study period indicating the stability of the local populations. The average density of the Common Raven was 1.3 birds/km2 and the highest density was registered in small villages (2.2 birds/km2), the lowest – in the townships (small urban type settlements) (0.6 birds/km2). The urban-rural gradient significantly affected the spatial distribution of Common Ravens, but we cannot claim an increase or decrease in their numbers along the gradient of urbanization, since the value of this indicator also depended on habitat conditions in each specific settlement. The spatial distribution of Common Ravens varied seasonally and the highest density was typical in winter due to food migrations towards human settlements.

Computation methods for combinatorial sums and Euler‐type numbers related to new families of numbers

The aim of this article is to define some new families of the special numbers. These numbers provide some further motivation for computation of combinatorial sums involving binomial coefficients and the Euler kind numbers of negative order. We can show that these numbers are related to the well‐known numbers and polynomials such as the Stirling numbers of the second kind and the central factorial numbers, the array polynomials, the rook numbers and polynomials, the Bernstein basis functions and others. In order to derive our new identities and relations for these numbers, we use a technique including the generating functions and functional equations. Finally, we give not only a computational algorithm for these numbers but also some numerical values of these numbers and the Euler numbers of negative order with tables. We also give some combinatorial interpretations of our new numbers. Copyright © 2016 John Wiley & Sons, Ltd.

An elliptic extension of the general product formula for augmented rook boards

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi, and White in 1975 showed that the sum over k of the product of the (n−k)th rook numbers multiplied by the kth falling factorial polynomials factorize into a product. In the sequel, different types of generalizations and analogues of this product formula have been derived by various authors. In 2008, Miceli and Remmel constructed a rook theory model involving augmented rook boards in which they showed the validity of a general product formula which can be specialized to all other product formulas that so far have appeared in the literature on rook theory. In this work, we construct an elliptic extension of the q-analogue of Miceli and Remmel’s result. Special cases yield elliptic extensions of various known rook theory models.

Bijections on [formula omitted]-level rook placements

Suppose the rows of a board are partitioned into sets of m rows called levels. An m-level rook placement is a subset of the board where no two squares are in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of m-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. This bijection also preserves the m-inversion number statistic of an m-level rook placement, defined by Briggs and Remmel. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards, but it yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne.

Spatial distribution of Corvidae in transformed landscapes of Zhytomyr region

The spatial distribution and abundance of Corvidae species was studied in Zhytomyr region with a focus on rural and urban differences in the studied parameters. We selected Rook (Corvus frugilegus L.), Western Jackdaw (C. monedula L.), Hooded Crow (C. cornix L.), Eurasian Magpie (Pica pica L.), Eurasian Jay (Garrulus glandarius L.), and Common Raven (Corvus corax L.). All observations were made during 2009–2012. During the study period some 38 survey paths of more than 8,000 km were surveyed in 21 settlements of Zhytomyr region, among them 13 were in Zhytomyr city. The aim of our study was to establish the number and density of Corvidae in different seasons in the settlements of Zhytomyr region along a rural-urban gradient. The average density of Rooks was 55.9 ind./km2. We also found a strong correlation between Rook density and the rural-urban gradient and observed that the number of Rooks wintering in cities significantly increased due to the influx from villages. The peak number of Rooks in villages was registered in the breeding and post-breeding season while in the cities it was high in winter and during the spring migration. The average density of Eurasian Magpie in the study area was 8.7 ind./km2 and had a weak correlation with the urban-rural gradient. The density of Eurasian Magpies in urban areas differs significantly only from the density of birds in villages with a population of ca. 1,000 people. The density of Magpies varied insignificantly within a narrow range during the three years of research, remaining relatively stable, which suggests that the species successfully adjusts to conditions in transformed landscapes. The urban-rural gradient significantly affects the density of Hooded Crows. The average density of birds in towns was 6.6 ind./km2. In breeding period the urban birds had a low density and rural crows, on the contrary, had a high density, the density of birds in the nesting period was greater than in autumn and winter, due to the distribution of birds in their breeding territories and the start of the reproductive cycle. The average density of Eurasian Jackdaw was 9.7 ind./km2 and the rural-urban gradient significantly affected the number of birds. For jackdaws we registered high annual population dynamics. The peak number of birds was observed in the winter period in cities and small towns while in villages it was registered in the breeding and post-breeding season. The average density of Eurasian Jay was 2.4 ind./km2 and varied significantly in different settlements, although the fluctuation range was small (0.1–9.3 ind./km2). The number of this species naturally decreases along the gradient of landscape transformation. The highest average density was typical in small villages (4.9 ind./km2) and the lowest was found in medium-sized cities (1.5 ind./km2). We have identified certain patterns in the seasonal dynamic of Eurasian Jays relative to the type of settlement. In urban areas the birds’ density was much higher in winter compared to the nesting period, and in villages we observed the reverse situation. The average density of Common Raven was 1.3 ind./km2. The highest density was registered in small villages and was 2.2 ind./km2, and the smallest in the villages of urban type (0.6 ind./km2). The urban-rural gradient significantly affected the spatial distribution of Common Raven, but we cannot state that this bird increased or decreased its number along the gradient of urbanization, since the value of this indicator also depended on habitat conditions in specific settlements. The spatial distribution of Common Ravens varied seasonally and the highest density was typical in winter due to food migrations to human settlements.

M-Level rook placements

Goldman, Joichi, and White proved a beautiful theorem showing that the falling factorial generating function for the rook numbers of a Ferrers board factors over the integers. Briggs and Remmel studied an analogue of rook placements where rows are replaced by sets of m rows called levels. They proved a version of the factorization theorem in that setting, but only for certain Ferrers boards. We generalize this result to any Ferrers board as well as giving a p,q-analogue. We also consider a dual situation involving weighted file placements which permit more than one rook in the same row. In both settings, we discuss properties of the resulting equivalence classes such as the number of elements in a class. In addition, we prove analogues of a theorem of Foata and Schützenberger giving a distinguished representative in each class as well as make connections with the q,t-Catalan numbers. We end with some open questions raised by this work.

Bijections on m-level Rook Placements

Partition the rows of a board into sets of $m$ rows called levels. An $m$-level rook placement is a subset of squares of the board with no two in the same column or the same level. We construct explicit bijections to prove three theorems about such placements. We start with two bijections between Ferrers boards having the same number of $m$-level rook placements. The first generalizes a map by Foata and Schützenberger and our proof applies to any Ferrers board. The second generalizes work of Loehr and Remmel. This construction only works for a special class of Ferrers boards but also yields a formula for calculating the rook numbers of these boards in terms of elementary symmetric functions. Finally we generalize another result of Loehr and Remmel giving a bijection between boards with the same hit numbers. The second and third bijections involve the Involution Principle of Garsia and Milne. Nous considérons les rangs d’un échiquier partagés en ensembles de $m$ rangs appelés les niveaux. Un $m$-placement des tours est un sous-ensemble des carrés du plateau tel qu’il n’y a pas deux carrés dans la même colonne ou dans le même niveau. Nous construisons deux bijections explicites entre des plateaux de Ferrers ayant les mêmes nombres de $m$-placements. La première est une généralisation d’une fonction de Foata et Schützenberger et notre démonstration est pour n’importe quels plateaux de Ferrers. La deuxième généralise une bijection de Loehr et Remmel. Cette construction marche seulement pour des plateaux particuliers, mais ça donne une formule pour le nombre de $m$-placements en terme des fonctions symétriques élémentaires. Enfin, nous généralisons un autre résultat de Loehr et Remmel donnant une bijection entre deux plateaux ayant les mêmes nombres de coups. Les deux dernières bijections utilisent le Principe des Involutions de Garsia et Milne.

On a New Family of Generalized Stirling and Bell Numbers

A new family of generalized Stirling and Bell numbers is introduced by considering powers $(VU)^n$ of the noncommuting variables $U,V$ satisfying $UV=VU+hV^s$. The case $s=0$ (and $h=1$) corresponds to the conventional Stirling numbers of second kind and Bell numbers. For these generalized Stirling numbers, the recursion relation is given and explicit expressions are derived. Furthermore, they are shown to be connection coefficients and a combinatorial interpretation in terms of statistics is given. It is also shown that these Stirling numbers can be interpreted as $s$-rook numbers introduced by Goldman and Haglund. For the associated generalized Bell numbers, the recursion relation as well as a closed form for the exponential generating function is derived. Furthermore, an analogue of Dobinski's formula is given for these Bell numbers.

MODELING HABITAT SELECTION OF THE RED-FOOTED FALCON ( FALCO VESPERTINUS ): A POSSIBLE EXPLANATION OF RECENT CHANGES IN BREEDING RANGE WITHIN HUNGARY

Due to a severe population decline and shrinkage of distribution range in the past decades, the redfooted falcon has gained top prior ity in both worldwide and Hungarian nature conservation. As a facultative colonial breeder, in Hungary, this species predominantly nests in rookeries. The number of rooks (Corvus frugilegus ) has also dramatically fallen recently, but population decline did not affect the large scale breeding distribution of this species. In our study we analyzed the presence of red� footed falcons at a colony in the case of current and historical breeding ranges based on landscape scaled habitat variables. We used a potential colony homerange size, estimated from observed homerange sizes in order t o determine the scale of influential habitat variables. According to our results, the primary cause of the observed range shift is the urbanization of rooks in definable regions of Hungary. The ratio of forests and open water surfaces within the potential homerange had negative, while the ratio of grasslands had a positive effect on the probability of redfooted falcon presence. None of our models predicted redfooted falcon presence at colonies outside the current breeding range, suggesting that a probable increase in redfooted falcon population numbers will not be accompanied by the expansion of the current breeding range.