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Parts~I--IV showed that the number of ways to place $q$ nonattacking queens or similar chess pieces on an $n\times n$ chessboard is a quasipolynomial function of $n$ whose coefficients are essentially polynomials in $q$. For partial queens, which have a subset of the queen's moves, we proved complete formulas for these counting quasipolynomials for small numbers of pieces and other formulas for high-order coefficients of the general counting quasipolynomials. We found some upper and lower bounds for the periods of those quasipolynomials by calculating explicit denominators of vertices of the inside-out polytope. Here we discover more about the counting quasipolynomials for partial queens, both familiar and strange, and the nightrider and its subpieces, and we compare our results to the empirical formulas found by \Kot. We prove some of \Kot's formulas and conjectures about the quasipolynomials and their high-order coefficients, and in some instances go beyond them.
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참고문헌 (16)
참고문헌 신청
S. E. Arshon / 1936 / Reshenie odnokombinatorornozadachi / Mat. Prosveshchenie 8:24~29
M. Beck / 2006 / Inside-out polytopes / Adv. Math 205(1):134~162
S. Chaiken / 2014 / A q-queens problem. I. General the-ory / Electron. J. Combin 21(3):28
S. Chaiken / 2015 / A q-queens problem. II. The square board / J. Algebraic Combin 41(3):619~642
S. Chaiken / 2019 / A q-queens problem III. Nonattacking partial queens / Australas. J. Combin 74:305~331
M. Beck / 2006 / Inside-out polytopes / Adv. Math 205(1):134~162
S. Chaiken / 2014 / A q-queens problem. I. General the-ory / Electron. J. Combin 21(3):28
S. Chaiken / 2015 / A q-queens problem. II. The square board / J. Algebraic Combin 41(3):619~642
S. Chaiken / 2019 / A q-queens problem III. Nonattacking partial queens / Australas. J. Combin 74:305~331
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