2,646 research outputs found
The distribution and abundance of the rook corvus frugilegus L. as influenced by habitat suitability and competitive interactions.
No full textRooks (Corvus frugilegus) are colonially breeding corvids found in most agricultural landscapes. Colonies in the County Durham area tend to be clustered at distances up to 500 m, but otherwise show little pattern in terms of spacing or size. Colony size was comparable between sites as changes in colony nest counts were allowed to stabilise before the whole area was surveyed. When measuring nest build-up at a sample of colonies in 1996, no further significant increases occurred after 9th April. The spatial size distribution of colonies was maintained between years. The distribution and size of breeding colonies is modelled in relation to the interaction between the spatial distribution of the foraging habitat and potential intraspecific competitors, with the identification of the distance over which this interaction is strongest. The satellite derived habitat data used for the modelling were part of the ITE Land Cover Map of Great Britain. However, their correspondence with ground reference data was found to be severely lacking. Thus, for modelling the availability of nesting habitat, OS woodland data were used as these identified more of the extant rookery sites, whilst the ITE data were retained for quantifying the foraging habitat. Logistic regression showed that the distribution of colony sites was influenced by the availability of woodland blocks large enough to hold a colony, proximity to roads and buildings, and by the amount of pasture within 1 km. Other suitable sites with these characteristics remained unoccupied within the distribution. Partial Correlations showed that interactions between the spatial distribution of the foraging habitat and competitors influenced colony size at distances up to 6 km, suggesting their effect outside of the breeding season. The multiple regression model built with variable values for this distance explained 31% of the variance in colony size. When applied to the potential breeding sites identified using the logistic regression, most sites still remained suitable. This suggests the distribution is not saturated and that limited availability of breeding habitat is not the cause of the nesting aggregations. The broad correlation of Rook abundance to foraging habitat and potential competitors corresponds to an ideal free distribution of individuals across colony sites. This is supported by models of Rook numbers in relation to parish agricultural statistics produced by MAFF. These again show the importance of pasture as a probable foraging resource, and how pasture quality could be important to Rook numbers. The models also supported the ideal free predictions of spatial variation in Rook abundance in relation to habitat, and the response of colony sizes to temporal change in habitat quality
M-level rook placements
No full textRook theory focuses on placements of non-attacking rooks on boards of various shapes. An important role is played by the rook numbers which count the number of non-attacking placements of a given number of rooks on a board. Ferrers boards,which are boards indexed by integer partitions, are of particular interest. Briggs and Remmel introduced a generalization of rook placements, called m-level rook placements, where a rook is able to attack a subset of the rows.This manuscript presents generalizations of many of the central results regarding rook placements to the case of m-level rook placements. Goldman, Joichi, and White defined the rook polynomial of a board to be the generating function for the rook numbers of that board in the falling factorial basis. By doing so, they were able to give an elegant factorization of the rook polynomial of a Ferrers board in terms of the various column heights. Briggs and Remmel were able to generalize this factorization to the m-level rook polynomial of a subset of Ferrers boards called singleton boards.We give two factorization theorems for the m-level rook polynomial of a Ferrers board. The first is a generalization of the factorization theorem of Briggs and Remmel, working from similar principles. The second relies on a generalization of transposition which we present, called the l-operator. We are also able to use the factorization to describe a unique representative in any m-level equivalence class of Ferrers boards and count the number of singleton boards in the class..When generalizing the factorization from singleton boards to all Ferrers boards, we preserve the definition of the m-level rook polynomial and alter the factorization to apply to all Ferrers boards. We also consider the dual of this problem, applying the factorization of Briggs and Remmel to all Ferrers boards, then trying to determine what is counted by the coefficients of the polynomial in the m-falling factorial basis. It turns out that the coefficients count weighted file placements on a Ferrers board. We also describe a unique representative in each weighted file placement equivalence class of Ferrers boards, as well as count of the number of Ferrers boards in a given weighted file placement equivalence class.Foata and Sch\ufc}tzenberger presented explicit bijections between rook placements on any two rook equivalent Ferrers boards as part of their construction of a unique representative in each equivalence class of Ferrers boards. A key tool in their construction was local transposition. We present analogous bijections between m-level rook placements on any two -level rook equivalent Ferrers boards using the local l-operator.The Garsia-Milne Involution Principle was first used in Garsia and Milne's bijective proof of the Rogers-Ramanujan identities. We use it to construct two types of explicit bijections. The first is an explicit bijection between m-level rook placements on any two m-level rook equivalent singleton boards. The second bijection is between the sets counted by the m-level analogue of hit numbers of any two m-level rook equivalent Ferrers boards, providing a bijective proof that -level equivalent Ferrers boards have the same hit numbers.(Ph. D.)--Michigan State University. Mathematics - Doctor of Philosophy, 2015Includes bibliographical reference
Rook polynomials
No full text碩士國際象棋中的車可直行與橫行。如果在任意形狀的棋盤上放置數個車，使得這些車不互相攻擊，則每個車必須彼此位在不同行不同列上。車多項式是指將車放置在棋盤上的方法數之生成函數。車多項式可用來解決有限制的排列的問題。因此我們希望能藉由探討一些特殊國際象棋中的車可直行與橫行。如果在任意形狀的棋盤上放置數個車，使得這些車不互相攻擊，則每個車必須彼此位在不同行不同列上。車多項式是一種放置各種不同個數的車的方法數的生成函數。車多項式可用來解決有限制的排列的問題。因此我們希望能藉由探討一些特殊棋盤的車多項式，獲得更快速解決有限制的排列的問題。
在論文中，我們主要推導並證明了四種特殊棋盤的車多項式：
1.m×n棋盤的車多項式。
2.有禁區的車多項式。
3.路徑棋盤的車多項式。
4.迴圈棋盤的車多項式。In combinatorial mathematics, a rook polynomial is a generating function of the number of ways to place non-attacking rooks on a board that looks like a checker board; that is, no two rooks can be placed in the same row or same column. The term "rook polynomial" was coined by John Riordan. Despite the name''s derivation from chess, the impetus for studying rook polynomials is their connection with counting the number of permutations with restricted positions.
In this thesis, we mainly obtain the rook polynomials of four special boards:
1.The rook polynomial of m×n chess board.
2.The rook polynomial with restricted area
3.The rook polynomial of path chess board
4.The rook polynomial of cycle chess board第一章 簡介 1
第二章 預備知識 3
2.1 排列組合 3
2.1.1 排列 3
2.1.2 組合 4
2.2 西洋棋(國際象棋) 4
2.3 車多項式 6
2.4特殊子棋盤 8
2.5 基本定理 9
第三章 主要結果 13
3.1棋盤的車多項式 13
3.2 特殊子棋盤的車多項式 15
3.3有禁區之子棋盤的車多項式 29
參考文獻 33
圖 目 錄
圖(1) 3×3棋盤 7
圖(2) 3×3 棋盤，放置1個車 7
圖(3) 3×3 棋盤，放置2個車 7
圖(4) 3×3 棋盤，放置3個車 7
圖(5) 8×8棋盤 8
圖(6) m×n棋盤 8
圖(7) m×n棋盤，對角線是禁區 8
圖(8) 1×n棋盤 8
圖(9) L(m,n) 8
圖(10) P(2,n) 8
圖(11) C(2,n) 9
圖(12) S''n 9
圖(13) L(1) 9
圖(14) L(2) 9
圖(15) L(2,2) 9
圖(16) L(3,3) 9
圖(17) 〖 T=T〗_1 ⋃▒T_(2 ) 10
圖(18)子棋盤 T 10
圖(19) m×n棋盤 13
圖(20)子棋盤 T 15
圖(21) P(2,n) 16
圖(22)P(2,n)利用定理2.5.2降階 18
圖(23) P(2,1)、P(2,2) 19
圖(24) C(2,n) 19
圖(25) C(2,n) 20
圖(26) C(2,4)、C(2,6) 20
圖(27) P(3,n) 21
圖(28) P(3,3k-2) 21
圖(29) P(3,3k-1) 21
圖(30) P(3,3k) 22
圖(31) P(k,n) 22
圖(32) P(k,mk-(k-1)) 23
圖(33) P(k,mk-(k-2)) 23
圖(34) P(k,mk-(k-3)) 23
圖(35) P(k,mk-(k-4) 23
圖(36) P(k,mk-(k-5) 24
圖(37) S''(n) 25
圖(38) S''(n) 25
圖(39) S''(10) 26
圖(40) P(5,25,3) 27
圖(41) 化簡後其規律性消失 28
圖(42) A(x) 28
圖(43) A(x) 30
圖(44) A(x) 31
圖(45) A(x) 31
表 目 錄
表(1) S(n,k) 的值 12
表(2) S(n,k)的值 27學號: 799190136, 學年度: 10
Applicazioni di paleontologia virtuale su resti fossili e blocchi ossiferi di Cava Monticino (Brisighella, RA)
No full textBijections on m-level Rook Placements
No full textPartition 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. Résumé. 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
La fauna di Cava Monticino (Brisighella, RA) nel contesto dei popolamenti continentali dell’area circum-mediterranea durante il Miocene terminale
No full textLa fauna di Cava Monticino (Brisighella, RA) nel contesto dei popolamenti continentali dell’area circum-mediterranea durante il Miocene terminale
No full textApplicazioni di paleontologia virtuale su resti fossili e blocchi ossiferi di Cava Monticino (Brisighella, RA)
No full textPositroidal aspects of non-nesting rook placements
No full textRook matroids were recently introduced by the author and Alexandersson as matroids whose bases arise from certain restricted rook placements on a skew-shaped board. They were shown to be a subclass of transversal matroids and positroids. We further investigate the structural properties of rook matroids with an emphasis on the positroidal point of view. In particular, we characterize rook matroids in terms of Grassmann necklaces of positroids, answering a question of Lam (2024). Along the way, we give a new proof of the positroidal structure of rook matroids and determine an important subclass of their cyclic flats. QC 20250512</p
Positroidal aspects of non-nesting rook placements
No full textRook matroids were recently introduced by the author and Alexandersson as matroids whose bases arise from certain restricted rook placements on a skew-shaped board. They were shown to be a subclass of transversal matroids and positroids. We further investigate the structural properties of rook matroids with an emphasis on the positroidal point of view. In particular, we characterize rook matroids in terms of Grassmann necklaces of positroids, answering a question of Lam (2024). Along the way, we give a new proof of the positroidal structure of rook matroids and determine an important subclass of their cyclic flats. QC 20250512</p
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