A002464 Hertzsprung's problem: ways to arrange n non-attacking kings on an n X n board, with 1 in each row and column. Also number of permutations of length n without rising or falling successions.
1, 1, 0, 0, 2, 14, 90, 646, 5242, 47622, 479306, 5296790, 63779034, 831283558, 11661506218, 175203184374, 2806878055610, 47767457130566, 860568917787402, 16362838542699862, 327460573946510746, 6880329406055690790, 151436547414562736234, 3484423186862152966838
Offset: 0
Examples
a(4) = 2: 2413, 3142. a(5) = 14 corresponds to these 14 permutations of length 5: 13524, 14253, 24135, 24153, 25314, 31425, 31524, 35142, 35241, 41352, 42513, 42531, 52413, 53142.
References
- W. Ahrens, Mathematische Unterhaltungen und Spiele. Teubner, Leipzig, Vol. 1, 3rd ed., 1921; Vol. 2, 2nd ed., 1918. See Vol. 1, p. 271.
- F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.40.
Links
- N. J. A. Sloane, Table of n, a(n) for n = 0..500
- M. Abramson and W. O. J. Moser, Permutations without rising or falling w-sequences, Ann. Math. Stat., 38 (1967), 1245-1254.
- W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig: B. G. Teubner, 1901.
- Another Roof, A Lifelong Mathematical Obsession, YouTube video, 2023.
- Art of Problem Solving Forum, Number of permutative
- Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the poset of King-Non-Attacking permutations, arXiv:1905.02387 [math.CO], 2019.
- Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Counting King Permutations on the Cylinder, arXiv:2001.02948 [math.CO], 2020.
- Christoph Bärligea, On the dimension of the Fomin-Kirillov algebra and related algebras, arXiv:2001.04597 [math.QA], 2020.
- Robert Brignall, Vít Jelínek, Jan Kynčl, and David Marchant, Zeros of the Möbius function of permutations, arXiv:1810.05449 [math.CO], 2018.
- Doug Chatham, Independence and domination on shogiboard graphs, Recreational Mathematics Magazine 4.8 (2017), pp. 25-37.
- Doug Chatham, Reflections on the n+k dragon kings problem, Games and Puzzles, Recreational Mathematics Magazine (2018) 5.10, 39-55.
- Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).
- P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 373.
- Sela Fried, Further results on staircase (cyclic) words, arXiv:2501.11991 [math.CO], 2025. See p. 12.
- Sela Fried, Toufik Mansour, and Mark Shattuck, Counting k-ary words by number of adjacency differences of a prescribed size, arXiv:2504.03013 [math.CO], 2025. See p. 2.
- Michael Han, Tanya Khovanova, Ella Kim, Evin Liang, Miriam Lubashev, Oleg Polin, Vaibhav Rastogi, Benjamin Taycher, Ada Tsui and Cindy Wei, Fun with Latin Squares, arXiv:2109.01530 [math.HO], 2021.
- C. Homberger, Counting Fixed Length Permutation Patterns, Online Journal of Analytic Combinatorics, 7 (2012).
- Cheyne Homberger, Patterns in Permutations and Involutions: A Structural and Enumerative Approach, arXiv preprint 1410.2657 [math.CO], 2014.
- Mark Huibregtse, Cristobal Lemus-Vidales, and David Vella, Bootstrap Percolation, Indecomposable Permutations, and the n-Kings problem, arXiv:2508.02030 [math.CO], 2025. See p. 2.
- Irving Kaplansky, The asymptotic distribution of runs of consecutive elements, Ann. Math. Statistics, 16 (1945), 200-203
- Sergey Kitaev and Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], (2013).
- Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, pp. 625-635.
- Igor Minevich, Gabe Cunningham, Aditya Karan, and Joshua V. Gyllinsky, Parks: A Doubly Infinite Family of NP-Complete Puzzles and Generalizations of A002464, arXiv:2411.02251 [cs.CC], 2024.
- O. Patashnik and P. Flajolet, Email to N. J. A. Sloane, Nov 26 1989
- P. Poulet, Query 4750: Permutations, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Page 117)
- P. Poulet, Query 4750: Permutations, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Pages 118, 119)
- P. Poulet, Query 4750: Permutations, L'Intermédiaire des Mathématiciens, 26 (1919), 117-121. (Pages 120, 121)
- J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.
- D. P. Robbins, The probability that neighbors remain neighbors after random rearrangements, Amer. Math. Monthly 87 (1980), 122-124.
- Josef Rukavicka, Secretary problem and two almost the same consecutive applicants, arXiv:2106.11244 [math.PR], 2021.
- A. J. Schwenk, Letter to N. J. A. Sloane, Mar 24 1980 (with enclosure and notes)
- L. Shapiro and A. B. Stephens, Bootstrap percolation, the Schroeder problems and the n-kings problem, SIAM J. Discrete Math., 4 (1991), 275-280.
- Jason P. Smith, A Formula for the Mobius function of the Permutation Poset Based on a Topological Decomposition, arXiv preprint arXiv:1506.04406 [math.CO], 2015.
- Roberto Tauraso, The Dinner Table Problem: The Rectangular Case, INTEGERS: Electronic Journal of Combinatorial Number Theory, Vol. 6 (2006), #A11. See Column 3 in the table on page 3.
- B. E. Tenner, Mesh patterns with superfluous mesh, arXiv preprint arXiv:1302.1883 [math.CO], 2013.
- Eric Weisstein's World of Mathematics, Hamiltonian Path
- Eric Weisstein's World of Mathematics, Path Complement Graph
- Eric Weisstein's World of Mathematics, Permutation
Crossrefs
Programs
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Maple
A002464 := proc(n) options remember; if n <= 1 then 1 elif n <= 3 then 0 else (n+1)*A002464(n-1)-(n-2)*A002464(n-2)-(n-5)*A002464(n-3)+(n-3)*A002464(n-4); fi; end;
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Mathematica
(* computation from the permutation class *) g[ L_ ] := Apply[ And, Map[ #>1&, L ] ]; f[ n_ ] := Length[ Select[ Permutations[ Range[ n ] ], g[ Rest[ Abs[ RotateRight[ # ]-# ] ] ]& ] ]; Table[ f[ n ], {n, 1, 8} ] (* Erich Friedman *) (* or direct computation of terms *) Table[n! + Sum[(-1)^r*(n-r)!*Sum[2^c *Binomial[r-1,c-1] *Binomial[n-r,c], {c,1,r}], {r,1,n-1}], {n,1,30}] (* Vaclav Kotesovec, Mar 28 2011 *) (* or from g.f. *) M = 30; CoefficientList[Sum[n!*x^n*(1-x)^n/(1+x)^n, {n, 0, M}] + O[x]^M, x] (* Jean-François Alcover, Jul 07 2015 *) CoefficientList[Series[Exp[(1 + x)/((-1 + x) x)] (1 + x) Gamma[0, (1 + x)/((-1 + x) x)]/((-1 + x) x), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *) RecurrenceTable[{a[n] == (n + 1) a[n - 1] - (n - 2) a[n - 2] - (n - 5) a[n - 3] + (n - 3) a[n - 4], a[0] == a[1] == 1, a[2] == a[3] == 0}, a, {n, 0, 20}] (* Eric W. Weisstein, Apr 11 2018 *) -
PARI
N = 66; x = 'x + O('x^N); gf = sum(n=0,N, n!*(x*(1-x))^n/(1+x)^n ); v = Vec(gf) /* Joerg Arndt, Apr 17 2013 */ -
Python
from math import factorial, comb def A002464(n): return factorial(n)+sum((-1 if k&1 else 1)*factorial(n-k)*sum(comb(k-1,t-1)*comb(n-k,t)<
Chai Wah Wu, Feb 19 2024
Formula
If n = 0 or 1 then a(n) = 1; if n = 2 or 3 then a(n) = 0; otherwise a(n) = (n+1)*a(n-1) - (n-2)*a(n-2) - (n-5)*a(n-3) + (n-3)*a(n-4).
G.f.: Sum_{n >= 0} n!*x^n*(1-x)^n/(1+x)^n. - Philippe Flajolet
G.f.: e^((1 + x)/((-1 + x) * x)) * (1 + x) * Gamma(0, (1 + x)/((-1 + x) * x))/((-1 + x) * x). - Eric W. Weisstein, May 16 2014
Let S_{n, k} = number of permutations of 12...n with exactly k rising or falling successions. Let S[n](t) = Sum_{k >= 0} S_{n, k}*t^k. Then S[0] = 1; S[1] = 1; S[2] = 2*t; S[3] = 4*t+2*t^2; for n >= 4, S[n] = (n+1-t)*S[n-1] - (1-t)*(n-2+3*t)*S[n-2] - (1-t)^2*(n-5+t)*S[n-3] + (1-t)^3*(n-3)*S[n-4].
a(n) = n! + Sum_{k=1..n} (-1)^k * Sum_{t=1..k} binomial(k-1,t-1) * binomial(n-k,t) * 2^t * (n-k)!. - Max Alekseyev, Jan 29 2006
a(n) = Sum_{k=0..n} (-1)^(n-k)*k!*b(n,k), where g.f. for b(n,k) is (1-x)/(1-(1+y)*x-y*x^2), cf. A035607. - Vladeta Jovovic, Nov 24 2007
Asymptotic (M. Abramson and W. Moser, 1966): a(n)/n! ~ (1 - 2/n^2 - 10/(3*n^3) - 6/n^4 - 154/(15*n^5) - 88/(9*n^6) + 5336/(105*n^7) + 1612/(3*n^8) + 2098234/(567*n^9) + 36500686/(1575*n^10) + ... )/e^2. - Vaclav Kotesovec, Apr 19 2011, extended Dec 27 2020
Conjecture: a(n) = Sum_{k=1..n} k!*A080246(n-1, k-1) for n > 0. - John Keith, Nov 02 2020
Proof: a(n) = Sum_{k=1..n} k!*A080246(n-1, k-1) for n > 0. Since a(n) = Sum_{k=0..n-1} (-1)^k*(n-k)!*Sum_{i=0..k} binomial(n-k,i)*binomial(n-1-i,k-i) (M. Abramson and W. Moser, 1966) which is Sum_{k=1..n} (-1)^(k-1)(n-k+1)!*Sum{i=0..k-1} binomial(n-k+1,i)*binomial(n-1-i,k-1-i) = Sum_{k=1..n} (-1)^(n-k)(k!)*Sum_{i=0..n-k} binomial(k,i)*binomial(n-1-i,n-k-i) = k!*A080246(n-1, k-1) as (-1)^(n-k) = (-1)^(n+k) and binomial(n-1-i,k-1) = binomial(n-1-i,n-k-i). - Alex McGaw, Apr 13 2023
a(n+2) = (n+2)! - Sum_{j=0..n} (-1)^j*(n+1-j)!*2*A104698(n, j), for n >= 0 (Abramson and Moser, p. 1250, (III), N_0(n+2), last line, rewritten). - Wolfdieter Lang, May 14 2025
Extensions
Merged with the old A001100, Aug 19 2003
Kaplansky reference from David Callan, Oct 29 2003
Tauraso reference from Parthasarathy Nambi, Dec 21 2006
Edited by Jon E. Schoenfield, Jan 31 2015
Comments