# Rooks Keep

When you were young, you thought the great walls of Barrow Keep would keep you safe. But now you're coming of age, and you realize how many troubles have been inside these walls all along: duplicitous courtiers, treacherous kin, and puritanical heresy-hunters. How will you protect yourself and your friends?

## Rooks Keep

Structurally, Barrow Keep is quite interesting in that it doesn't approach the game world like a sandbox but like notes for a novel. For example, the first thing you do when you crack it open is decide the names and identities of several story-pivotal characters. *Then* you get some information on the titular keep.

While there are some new items, types of magic, weapons, and other mechanical bits included in the book, its real meat is the keep's secrets. The second half of the book is adventures, hooks, factions, maps, and a whole host of ways for youngsters living in or near the keep to get into trouble.

Along those same lines, keep Nerlens Noel in the city. Resist the temptation of the cash or trade incentives that bring notoriety to the management at the press conference. Noel came at a time shortly thereafter the Sixers uniforms went back to their classic lettering and color ensemble. His game and unique abilities as a center are also a classic ensemble, especially for a revolutionizing team and coach.

I do the same thing. I give up a rook trade too easily, even though with two rooks in an endgame you have a big advantage. Must think of rooks like the game finishers called in the ninth inning to win it.

Skittish. Enemies will randomly drop all threat on their currenttarget. This makes it significantly more dangerous for DPS and adds an extrafactor for tanks to account for, as dangerous mobs will require additionalattention to keep in check.

Staff at the Puy du Fou park in the western Vendee region of France, have trained six rooks (Corvus frugilegus) named Boubou, Bamboo, Bill, Black, Bricole and Baco to pick up little bits of garbage off the ground and insert them into a box that automatically dispenses a treat.

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Rooks are most usually seen in flocks in open fields, or feeding in small groups along a roadside. They will come into town parks and villages but largely keep clear of the middle of big towns and cities. They are absent from the far north west of Scotland.

In combinatorial mathematics, a rook polynomial is a generating polynomial of the number of ways to place non-attacking rooks on a board that looks like a checkerboard; that is, no two rooks may be in the same row or column. The board is any subset of the squares of a rectangular board with m rows and n columns; we think of it as the squares in which one is allowed to put a rook. The board is the ordinary chessboard if all squares are allowed and m = n = 8 and a chessboard of any size if all squares are allowed and m = n. The coefficient of x k in the rook polynomial RB(x) is the number of ways k rooks, none of which attacks another, can be arranged in the squares of B. The rooks are arranged in such a way that there is no pair of rooks in the same row or column. In this sense, an arrangement is the positioning of rooks on a static, immovable board; the arrangement will not be different if the board is rotated or reflected while keeping the squares stationary. The polynomial also remains the same if rows are interchanged or columns are interchanged.

The term "rook polynomial" was coined by John Riordan.[1]Despite the name's derivation from chess, the impetus for studying rook polynomials is their connection with counting permutations (or partial permutations) with restricted positions. A board B that is a subset of the n n chessboard corresponds to permutations of n objects, which we may take to be the numbers 1, 2, ..., n, such that the number aj in the j-th position in the permutation must be the column number of an allowed square in row j of B. Famous examples include the number of ways to place n non-attacking rooks on:

Interest in rook placements arises in pure and applied combinatorics, group theory, number theory, and statistical physics. The particular value of rook polynomials comes from the utility of the generating function approach, and also from the fact that the zeroes of the rook polynomial of a board provide valuable information about its coefficients, i.e., the number of non-attacking placements of k rooks.

where r k ( B ) \displaystyle r_k(B) is the number of ways to place k non-attacking rooks on the board B. There is a maximum number of non-attacking rooks the board can hold; indeed, there cannot be more rooks than the number of rows or number of columns in the board (hence the limit min ( m , n ) \displaystyle \min(m,n) ).[2]

In words, this means that on a 1 1 board, 1 rook can be arranged in 1 way, and zero rooks can also be arranged in 1 way (empty board); on a complete 2 2 board, 2 rooks can be arranged in 2 ways (on the diagonals), 1 rook can be arranged in 4 ways, and zero rooks can be arranged in 1 way; and so forth for larger boards.

We deduce an important fact about the coefficients rk, which we recall given the number of non-attacking placements of k rooks in B: these numbers are unimodal, i.e., they increase to a maximum and then decrease. This follows (by a standard argument) from the theorem of Heilmann and Lieb[3] about the zeroes of a matching polynomial (a different one from that which corresponds to a rook polynomial, but equivalent to it under a change of variables), which implies that all the zeroes of a rook polynomial are negative real numbers.

A precursor to the rook polynomial is the classic "Eight rooks problem" by H. E. Dudeney[4] in which he shows that the maximum number of non-attacking rooks on a chessboard is eight by placing them on one of the main diagonals (Fig. 1). The question asked is: "In how many ways can eight rooks be placed on an 8 8 chessboard so that neither of them attacks the other?" The answer is: "Obviously there must be a rook in every row and every column. Starting with the bottom row, it is clear that the first rook can be put on any one of eight different squares (Fig. 1). Wherever it is placed, there is the option of seven squares for the second rook in the second row. Then there are six squares from which to select the third row, five in the fourth, and so on. Therefore the number of different ways must be 8 7 6 5 4 3 2 1 = 40,320" (that is, 8!, where "!" is the factorial).[5]

The same result can be obtained in a slightly different way. Let us endow each rook with a positional number, corresponding to the number of its rank, and assign it a name that corresponds to the name of its file. Thus, rook a1 has position 1 and name "a", rook b2 has position 2 and name "b", etc. Then let us order the rooks into an ordered list (sequence) by their positions. The diagram on Fig. 1 will then transform in the sequence (a,b,c,d,e,f,g,h). Placing any rook on another file would involve moving the rook that hitherto occupied the second file to the file, vacated by the first rook. For instance, if rook a1 is moved to "b" file, rook b2 must be moved to "a" file, and now they will become rook b1 and rook a2. The new sequence will become (b,a,c,d,e,f,g,h). In combinatorics, this operation is termed permutation, and the sequences, obtained as a result of the permutation, are permutations of the given sequence. The total number of permutations, containing 8 elements from a sequence of 8 elements is 8! (factorial of 8).

To assess the effect of the imposed limitation "rooks must not attack each other", consider the problem without such limitation. In how many ways can eight rooks be placed on an 8 8 chessboard? This will be the total number of combinations of 8 rooks on 64 squares:

The classical rooks problem immediately gives the value of r8, the coefficient in front of the highest order term of the rook polynomial. Indeed, its result is that 8 non-attacking rooks can be arranged on an 8 8 chessboard in r8 = 8! = 40320 ways.

Let us generalize this problem by considering an m n board, that is, a board with m ranks (rows) and n files (columns). The problem becomes: In how many ways can one arrange k rooks on an m n board in such a way that they do not attack each other?

It is clear that for the problem to be solvable, k must be less or equal to the smaller of the numbers m and n; otherwise one cannot avoid placing a pair of rooks on a rank or on a file. Let this condition be fulfilled. Then the arrangement of rooks can be carried out in two steps. First, choose the set of k ranks on which to place the rooks. Since the number of ranks is m, of which k must be chosen, this choice can be done in ( m k ) \displaystyle \binom mk ways. Similarly, the set of k files on which to place the rooks can be chosen in ( n k ) \displaystyle \binom nk ways. Because the choice of files does not depend on the choice of ranks, according to the products rule there are ( m k ) ( n k ) \displaystyle \binom mk\binom nk ways to choose the square on which to place the rook.

However, the task is not yet finished because k ranks and k files intersect in k2 squares. By deleting unused ranks and files and compacting the remaining ranks and files together, one obtains a new board of k ranks and k files. It was already shown that on such board k rooks can be arranged in k! ways (so that they do not attack each other). Therefore, the total number of possible non-attacking rooks arrangements is:[6] 041b061a72