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Mathematical articles and books by Vaclav Kotesovec

  • Asymptotics for a certain group of exponential generating functions - arXiv:2207.10568 [math.CO], 13 Jul 2022 - a combination of the Hayman's method, Newton's method and limits with LambertW function (in addition: bug in Mathematica found!). OEIS A143405, A355291, A002872, A002874.
  • Asymptotic formula for OEIS A032302 - a combination of the Euler-MacLaurin summation formula and the saddle point method. Formula (and generalization) for A032302 published Jan 04 2016, Mathematica notebook added Jun 21 2022
  • L-convex polyominoes and 201-avoiding ascent sequences, with Anthony Guttmann, arXiv:2109.09928 [math.CO], 21 Sep 2021
  • Asymptotics of sequence A002513, mirror - added 24.8.2019
  • A method of finding the asymptotics of q-series based on the convolution of generating functions - arXiv:1509.08708 [math.CO], 29 Sep 2015
  • Asymptotics of the Euler transform of Fibonacci numbers - arXiv:1508.01796 [math.CO], 7 Aug 2015
  • The partition factorial constant and asymptotics of the sequence A058694, mirror - added 26.6.2015
  • The integration of q-series, mirror - added 29.5.2015, updated 31.5.2015
  • Asymptotics of the sequence A120733, mirror - added 3.5.2015
  • Asymptotics of sequence A034691, mirror - application of the saddle-point method (added 9.9.2014)
  • Asymptotic of the coefficients A079330 / A088989, mirror - about roots of the equation tan(x)=x (added 20.8.2014)
  • Asymptotic solution of the equations using the Lambert W-function, mirror - method of solution of the equations, with an application of the theorem by Hayman (added 8.8.2014)
  • Asymptotic of subsequences of A212382, mirror - application of the theorem by Bender for ordinary generating functions (added 17.7.2014)
  • Asymptotic of sequences A161630, A212722, A212917 and A245265, mirror - application of the theorem by Bender for exponential generating functions (added 16.7.2014)
  • Asymptotic of sequences A244820, A244821 and A244822, mirror (added 11.7.2014)
  • Asymptotic of implicit functions if Fww = 0 - extension of theorem by Bender (added 19.1.2014)
  • Asymptotic of sequence A227403 - rigorous proof and conjecture (added 21.9.2013)
  • Asymptotic of Young tableaux of bounded height - conjecture (added 12.9.2013)
  • Asymptotic of sequence A084611 (added 26.7.2013)
  • Interesting asymptotic formulas for binomial sums (added 9.6.2013, extended 28.6.2013)
  • Asymptotic of generalized Apéry sequences with powers of binomial coefficients (added 4.11.2012, extended 23.11.2012)
  • Too many errors around coefficient C1 in asymptotic of sequence A002720 - I found bug in program Mathematica! (added 28.9.2012) Update 25.7.2015: This bug was fixed in version / July 2015.
  • Asymptotic of a sums of powers of binomial coefficients * x^k (added 20.9.2012)
  • Asymptotic formula for number of fat trees on n labeled vertices - OEIS A055779 (added 27.8.2012)

    Next over 16000 of my formulas, programs and comments can be found in the On-Line Encyclopedia of Integer Sequences, OEIS, User:Vaclav_Kotesovec (Editor-in-Chief)

    Non-attacking chess pieces (2013)

    A book is devoted to the question of the number of arrangements of non-attacking chess pieces of the same kind on chessboards of various sizes and types. The best-known example is the n-Queens problem, but this publication has a much wider range and includes other chess pieces (kings, rooks, bishops, knights) and many fairy pieces. Even though the book is about chess and each problem can be placed among chess-mathematical problems, it will be more readily understandable by mathematicians than by chess players or composers. A partial knowledge of linear algebra, difference equations, generating functions and power series is necessary.

      Non-attacking chess pieces (sixth edition)
    (chess and mathematics)

    by Vaclav Kotesovec

    795 pages, over 500 formulas, many tables

    published 2.2.2013 (minor update 8.9.2016)

    free download PDF (14 MB)

    New in the sixth edition (this edition is probably final):
  • For pieces Rookhopper and Bishophopper (include number of stalemate positions!) see new chapters 9.9
  • For maximal number of non-attacking riders [r,s], see updated chapter 14.1
  • New formulas for semi-knights and generally for semi-leapers, see new chapter 5.1.2
  • Enhanced table of entropy constants, see page 69
  • New recurrence for bishops on an toroidal chessboard n x n if n is even, see page 280
  • Both constants in the asymptotic formula for composite pieces semi-Rook + semi-Bishop are now in closed form!, see page 717
  • Formula for 10 non-attacking kings on an n x n chessboard, see updated chapter 2.1

    New in the fifth edition:
  • For general asymptotic formulas see new chapter 13.1
  • For new results for m x n non-attacking kings on a 2m x 2n chessboard see chapters 2.3 and 2.3.9
  • For non-attacking kings on the cylindrical chessboard, see the new chapters 2.6, 2.6.9, 2.7 and 2.5
  • For new formula for the number of ways of placing n2 non-attacking kings on an 2n x 2n toroidal chessboard, see the new chapter 2.9
  • For explicit formula for the number of ways of placing 6 non-attacking queens on an n x n toroidal chessboard, see the updated chapter 1.3
  • For non-attacking nightriders on the cylindrical chessboard, see the new chapters 6.5 and 6.6
  • Roots of a polynomials as points in the complex plane
  • Added pieces semi-wazir, semi-fers and semi-knight (interesting from mathematical view point), see new chapters 9.1.1, 9.3.1, 9.4.1 and 5.1.1
  • Reduced PDF size
    Errata 5th edition

    New in the fourth edition:
  • For general formulas for the number of ways of placing k non-attacking bishops on an n x n chessboard (including the most interesting case k=n), see the updated chapters 4.1, 4.3 and new chapters 4.1.1, 4.1.2., 4.4.
  • For miscellaneous problems with rooks, see chapter 3.1.1
  • An extensive new chapter 12 is devoted to the asymptotic behaviour of sequences of numbers of ways of placing non-attacking composite pieces rook + leaper, queen + leaper, rook + rider, queen + rider. There are many new formulas, conjectures, graphs and tables of values.
  • For maximal number of non-attacking pieces, see chapter 14

    New in the third edition:
  • 8n nonattacking kings on a 16 x 2n chessboard, formula for smallest root (include limit)
  • Formula for 5 non-attacking amazons on a n x n board
  • Kings, amazons and zebras on a toroidal chessboard
  • 65 new explicit formulas added to section Riders, total more than 300 explicit formulas in third edition now
  • New results and tables for 2 riders on a toroidal chessboard
  • Method for transformation of formulas with Floor function to expressions with trigonometric functions
  • Index of citations

    New in the second edition:
  • General recurrence for number of ways of placing k non-attacking queens on an n x n chessboard
  • Leapers
  • Riders

  • Number of ways of placing non-attacking queens, kings, bishops and knights on boards of various sizes - old article, website, 1996-2010

    Related articles:
  • A q-Queens Problem, I. General theory - The Electronic Journal of Combinatorics, Volume 21, Issue 3 (2014), Paper #P3.33, arXiv 1303.1879 [math.CO] - Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, 08 Mar 2013, updated 20 Feb 2014,
  • A q-Queens Problem. II. The square board - Journal of Algebraic Combinatorics, 41 (2015), no. 3, 619-642, arXiv 1402.4880 [math.CO] - Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, 20 Feb 2014, updated 07 Aug 2014
  • A q-Queens Problem. III. Nonattacking Partial Queens, The Australasian Journal of Combinatorics, volume 74(2) (2019), pages 305-331, arXiv 1402.4886 [math.CO] - Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, 20 Feb 2014
  • A q-Queens Problem. IV. Attacking Configurations and Their Denominators - Discrete Mathematics, 343(2) (2020), article 111649, Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, arXiv 1807.04741 [math.CO], 10 Jul 2018
  • A q-Queens Problem. V. Some of Our Favorite Pieces: Queens, Bishops, Rooks, and Nightriders - Journal of the Korean Mathematical Society 2020 Vol. 57, No. 6, 1407-1433, arXiv 1609.00853 [math.CO] - Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, 03 Sep 2016, last revised 17 Jul 2020.
  • A q-Queens Problem. VI. The Bishops' Period - Ars Mathematica Contemporanea 16 (2019), 549–561, arXiv 1405.3001 [math.CO] - Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, 12 May 2014
  • A q-Queens problem. VII. Combinatorial types of nonattacking chess riders - Australasian J. Combin., 77(3) (2020), 326–335, arXiv 1906.08981 [math.CO] - Christopher R.H. Hanusa, Thomas Zaslavsky, 21 Jun 2019
  • A q-Queens Problem - Christopher R. H. Hanusa (with Thomas Zaslavsky and Seth Chaiken), MOVES Conference, in New York City, August 2-4, 2015 (PDF / slides)
  • Nonattacking Queens in a Rectangular Strip - Seth Chaiken, Christopher R. H. Hanusa, Thomas Zaslavsky, Annals of Combinatorics 14 (2010), 419-441, arXiv:1105.5087 [math.CO], 25 May 2011
  • Non-attacking chess pieces: The dance of bishops - Thomas Zaslavsky, 29.7.2010
  • Podschet rasstanovok ferzey na shakhmatnoy doske - Artem M. Karavaev, Kazanskaya Nauka 10, 2010, p.13-16
  • Thomas Zaslavsky - Publications and So Forth
  • Roberto Tauraso - Papers - OEIS A078630
  • The q-Queens Problem: One-Move Riders on the Rectangular Board - Jaimal Ichharam, arXiv:1501.06642 [math.CO], 27 Jan 2015
  • Closed-Form Expressions for the n-Queens Problem and Related Problems - Kevin Pratt, International Mathematics Research Notices, Volume 2019, Issue 4, February 2019, Pages 1098–1107, arXiv 1609.09585 [math.CO], 2017.
  • The number of n-queens configurations - Michael Simkin, 2021.

    Other citations:
  • Mathematical Constants II - Steven R. Finch, 2019, ISBN: 9781108470599, p. 76 [OEIS A000294 and A255939], p. 267 [OEIS A000670, A032032, A102233, and A232475], p. 269 and 271 [OEIS A034691], p. 496 [OEIS A062980 and A094199]
  • Enumerative Combinatorics - Richard P. Stanley (vol. 1, 2ed, 2012, p.640 and 615, exercise 41).
  • The number of {1243, 2134}-avoiding permutations - David Callan, arXiv:1303.3857 [math.CO], 15 Mar 2013, OEIS A164651
  • Injections, Surjections and More - Steven Finch, 07 May 2015
  • Integer Partitions - Steven Finch, updated in 2015, p.5
  • Errata and Addenda to Mathematical Constants - Steven Finch, updated Jan 22 2016, p.42 and 117, OEIS A003242 and A032032
  • Errata and Addenda to Mathematical Constants - Steven Finch, arXiv:2001.00578 [math.HO], Jan 02 2020, p.43 and 123, OEIS A003242 and A032032
  • Mellin Transforms of the Generalized Fractional Integrals and Derivatives - Udita N. Katugampola, arXiv:1112.6031 [math.CA], 29 Oct 2014, OEIS A223168
  • Strong and ratio asymptotics for Laguerre polynomials revisited - Alfredo Deano, Edmundo J. Huertas, Francisco Marcellan, arXiv:1301.4266v2 [math.CA], 24 Jun 2013 (see Comments)
  • Two Vignettes On Full Rook Placements - J. Bloom, V. Vatter, arXiv:1310.6073 [math.CO], 22 Oct 2013, OEIS A165543
  • Two Vignettes On Full Rook Placements - J. Bloom, V. Vatter, The Australasian Journal of Combinatorics, vol. 64(1), 2016, p. 80, OEIS A165543
  • On a Mean Field Theory of Topological 2D Gravity - Jian Zhou, arXiv:1503.08546 [math.AG], 30 Mar 2015, OEIS A094199
  • Enumeration of standard Young tableaux of shifted strips with constant width - Ping Sun, arXiv:1506.07256 [math.CO], 24 Jun 2015, OEIS A181197
  • Polynomials, Quantum Query Complexity, and Grothendieck's Inequality - Scott Aaronson, Andris Ambainis, Janis Iraids, Martins Kokainis, Juris Smotrovs, arXiv:1511.08682 [quant-ph], 27 Nov 2015
  • Computationally Efficient Bounds for the Sum of Catalan Numbers - Kevin Topley, arXiv:1601.04223 [math.CO], 16 Jan 2016, OEIS A014138
  • Short note on the number of 1-ascents in dispersed dyck paths - Kairi Kangro, Mozhgan Pourmoradnasseri, Dirk Oliver Theis, arXiv:1603.01422 [math.CO], 04 Mar 2016, OEIS A191386
  • Efficient Counting of Degree Sequences or DOI - Kai Wang, arXiv:1604.04148 [math.CO], 14 Apr 2016, p. 2 and p. 13, OEIS A005155
  • Convergence criteria for FIW-algebras and polynomial statistics on maximal tori in type B/C - Rita Jimenez Rolland, Jennifer C. H. Wilson, arXiv:1604.06392 [math.CO], 21 Apr 2016, p. 14, OEIS A075196
  • Asymptotic enumeration by Khintchine-Meinardus method: Necessary and sufficient conditions for exponential growth - Boris Granovsky, arXiv:1606.08016 [math.CO], 26 Jun 2016, p. 2 and p. 29
  • Chained permutations and alternating sign matrices - inspired by three-person chess - Dylan Heuer, Chelsey Morrow, Ben Noteboom, Sara Solhjem, Jessica Striker, Corey Vorland, arXiv:1611.03387 [math.CO], 10 Nov 2016, p. 26 and p. 1
  • On the number of lattice convex chains - Julien Bureaux, Nathanaël Enriquez, arXiv:1603.09587 [math.PR], 31 Mar 2016 (updated 11 Dec 2016), p. 11 and p. 14, OEIS A267862
  • Some useful theorems for asymptotic formulas and their applications to skew plane partitions and cylindric partitions - Guo-Niu Han, Huan Xiong, arXiv:1707.04907 [math.CO], 16 Jul 2017, p. 1, 2 and 15. another version
  • Skew doubled shifted plane partitions: calculus and asymptotics - Guo-Niu Han, Huan Xiong, arXiv:1707.05832 [math.CO], 18 Jul 2017, p. 3, 10 and 17
  • k -Foldability of Words - B. Bjorkman, G. Cochran, W. Gao, L. Keough, R. Kirsch, M. Phillipson, D. Rorabaugh, H. Smith, J. Wise, arXiv:1710.10616 [math.CO], 29 Oct 2017, p. 13 (Acknowledgements)
  • A survey of known results and research areas for n-queens - Jordan Bell, Brett Stevens, Discrete Mathematics 309 (2009) 1–31, p. 20
  • Elementary Number Theory - Hatice Boylan and Nils-Peter Skoruppa, Lecture Notes Istanbul Universitesi and Universitat Siegen, 7/2016, p.85
  • On Certain Reciprocal Sums - Soumyadip Sahu, arXiv:1807.05454 [math.NT], 14 Jul 2018, p. 11, OEIS A248230, A248234
  • What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I - Martin Klazar, arXiv:1808.08449, Sep 11 2018, p.19, 38, 40, 46, 47, OEIS A000009, A000700, A292520, A002107, A258232.
  • A class of symmetric difference-closed sets related to commuting involutions - John M. Campbell, Discrete Mathematics and Theoretical Computer Science DMTCS vol. 19:1, 2017, #8, p.4, 7. OEIS A266503, A267840.
  • Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube - Yu Hin (Gary) Au, Fatemeh Bagherzadeh, Murray R. Bremner, arXiv:1903.00813, Mar 03 2019, p.10, 27. OEIS A236339. See also Journal of Integer Sequences, Vol. 23 (2020), Article 20.1.4.
  • On the asymptotic distinct prime partitions of integers - M. V. N. Murthy, M. Brack, R. K. Bhaduri, arXiv:1904.02776, Mar 22 2019, p. 10. OEIS A000586. New version May 10 2021, see p.10-11.
  • Exact finite-size corrections in the dimer model on a planar square lattice - Nikolay Sh. Izmailian, Vladimir V. Papoyan and Robert M. Ziff, Journal of Physics A: Mathematical and Theoretical, Jul 23 2019, p. 24, reference [48]. OEIS A012495.
  • Asymptotics and statistics on Fishburn matrices and their generalizations - Hsien-Kuei Hwang and Emma Yu Jin, arXiv:1911.06690 [math.CO], Nov 15 2019, p.33 OEIS A186737, p.35 OEIS A035378, p.36 OEIS A207557.
  • New results and conjectures on 2-partitions of multisets or DOI - Ovidiu Bagdasar and Dorin Andrica, 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO), 2017. OEIS A047653.
  • The r-alternating permutations, István Mező & José L. Ramírez, Aequationes mathematicae volume 94, pages 37–57 (2020). OEIS A225688.
  • Some inequalities for Garvan's bicrank function of 2-colored partitions - Shane Chern, Dazhao Tang, Liuquan Wang, arXiv:1805.06575 [math.CO], May 17 2018, p.3, p.27.
  • Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions - Wenjie Fang, Hsien-Kuei Hwang, Mihyun Kang, arXiv:2004.08901 [math.CO], Apr 19 2020, p.6 OEIS A266648, p.28 OEIS A225196-9.
  • Structure and enumeration of K4-minor-free links and link-diagrams - Juanjo Rué, Dimitrios M. Thilikos and Vasiliki Velona, arXiv:1806.07855 [math.CO], Apr 24 2020, p.17 (OEIS A022567) and p.29.
  • An introduction to the Bernoulli function - Peter H. N. Luschny, arXiv:2009.06743 [math.HO], Sep 11 2020, p.11 and p.33. HTML variant
  • Growth rate of 3D heaps of pieces - M.V. Tamm, N. Pospelov, S. Nechaev, arXiv:2009.12540 [cond-mat.stat-mech], Oct 02 2020, p.7-8 and 14.
  • On the largest Kronecker and Littlewood-Richardson coefficients - Igor Pak, Greta Panova, Damir Yeliussizov, arXiv:1804.04693 [math.CO], 2018, Remark 3.8, formula (3.5), p.8 - cited my asymptotic formula from OEIS A110143.
  • Asymptotics of descent functions - Kaarel Hänni, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14, OEIS A049774, A117158, A177523.
  • All-orders asymptotics of tensor model observables from symmetries of restricted partitions - Joseph Ben Geloun, Sanjaye Ramgoolam, arXiv:2106.01470 [hep-th], Jun 02 2021, p. 3, 5, 8, 19, 43. OEIS A110143, A279819.
  • Properties of Hamiltonian Circuits in Rectangular Grids - Rüdiger Jehn, arXiv:2103.15778 [math.GM], Mar 29 2021, p. 13, 16, OEIS A201629.
  • The Generalized Superfactorial, Hyperfactorial and Primorial functions - Vignesh Raman, arXiv:2012.00882 [math.NT], Dec 01 2020, p. 2, 17, OEIS A002109.
  • Bounds for Combinatorial Types of Non-Attacking Riders - Grant Jensen, arXiv:2006.12689 [math.CO], Jun 24 2020, p. 4, 5, 6.
  • Wilf Equivalences and Stanley-Wilf Limits for Patterns in Rooted Labeled Forests - Michael Ren, arXiv:2007.12690 [math.CO], Jul 27 2020, p. 48, OEIS A000262.
  • Dyson's Crank and the Mex of Integer Partitions - Brian Hopkins, James A. Sellers, Dennis Stanton, arXiv:2009.10873 [math.CO], Sep 23 2020, p. 4, OEIS A064410.
  • Quiddities of polygon dissections and the Conway-Coxeter frieze equation - Charles H. Conley and Valentin Ovsienko, arXiv:2107.01234 [math.CO], Jul 02 2021, p. 5, 6, 7, OEIS A218251.
  • Transportation Distance between Probability Measures on the Infinite Regular Tree - Pakawut Jiradilok and Supanat Kamtue, arXiv:2107.09876 [math.CO], Sep 18 2021, p.36, 37, OEIS A035610, A130976.
  • Closed-form formulas and properties of coefficients in Maclaurin's series expansion of Wilf's function - Feng Qi and Mark Daniel Ward, arXiv:2110.08576 [math.CO], Oct 16 2021, p. 20, 24, OEIS A014307, A180875.
  • Pattern-avoiding ascent sequences of length 3 - Andrew R. Conway, Miles Conway, Andrew Elvey Price and Anthony J. Guttmann, arXiv:2111.01279 [math.CO], Nov 01 2021, p. 3, 31, OEIS A202062.
  • Guessing with Little Data - Manuel Kauers and Christoph Koutschan, arXiv:2202.07966 [cs.SC], Feb 16 2022, p. 15, 17, OEIS A189281.
  • Counting Permutations Where The Difference Between Entries Located r Places Apart Can never be s (For any given positive integers r and s) - George Spahn and Doron Zeilberger, arXiv:2211.02550 [math.CO], Dec 06 2022, p. 2, 7, OEIS A189281, A189282, A189283, A189284, A189285 and A110128.
  • Maximum arrangements of nonattacking kings on the 2n×2n chessboard - Tricia Muldoon Brown, arXiv:2111.10331 [math.CO], Nov 19 2021, p. 2, 8, OEIS A018807.
  • Combinatorial Exploration: An algorithmic framework for enumeration - Michael H. Albert, Christian Bean, Anders Claesson, Émile Nadeau, Jay Pantone, Henning Ulfarsson, arXiv:2202.07715 [math.CO], Feb 15 2022, p. 77, 97.

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