A002493 -id:A002493 - cp's OEIS frontend

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

N. J. A. Sloane

Permutations of 12...n such that none of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
This sequence is also the solution to the 'toast problem' devised by my house-mates and me as math undergraduates some 27 years ago: Given a toast rack with n slots, how many ways can the slices be removed so that no two consecutive slices are removed from adjacent slots? - David Jones (david.jones(AT)zetnet.co.uk), Oct 24 2003
This sequence was also derived by the late D. P. Robbins. - David Callan, Nov 04 2003
Another interpretation: number of permutations of n containing exactly n different patterns of size n-1. - Olivier Gérard, Nov 05 2007
Number of directed Hamiltonian paths in the complement of the n-path graph P_n. - Andrew Howroyd, Mar 16 2016
There is an obvious connection between the two descriptions of the sequence: Replace the chessboard with a n X n zero-matrix and each king with "1". This matrix will transform the vector (1,2,..,n) into a permutation such that adjacent components do not differ by 1. The reverse is also true: Any such transformation is a solution of the king problem. - Gerhard Kirchner, Feb 10 2017
A formula of Poulet (1919) relates this to A326411: a(n) = T(n+2,1)/(n+2) + 2*T(n+1,1)/(n+1) + T(n,1)/n, where T(i,j) = A326411(i,j). - N. J. A. Sloane, Mar 08 2022
For the number of these permutations without fixed points see A288208. - Wolfdieter Lang, May 22 2025

			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.

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.

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

Equals 2*A001266(n) for n >= 2. A diagonal of A001100. Cf. A010028.
Cf. A086852, A086853, A086854, A086855, A000130, A000349, A001267, A001268, A010028, A002493.
Cf. A089222, A104698, A288208.
Column k=1 of A333706.

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;

(* 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 *)

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 */

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

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

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

A242522 Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at least 2.

Original entry on oeis.org

0, 0, 0, 0, 1, 5, 33, 245, 2053, 19137, 196705, 2212037, 27029085, 356723177, 5058388153, 76712450925, 1239124984693, 21241164552785, 385159565775633, 7365975246680597, 148182892455224845, 3128251523599365177, 69149857480654157545, 1597343462243140957757

Offset: 1

Stanislav Sykora, May 27 2014

nonn

a(n)=NPC(n;S;P) is the count of all neighbor-property cycles for a specific set S of n elements and a specific pair-property P. For more details, see the link and A242519.
Number of Hamiltonian cycles in the complement of P_n, where P_n is the n-path graph. - Andrew Howroyd, Mar 15 2016
a(n) also agrees with the number of optimal fundamentally distinct radio labelings of the wheel graph on (n+1) nodes for n = 5 up to at least n = 10 (and likely all larger n). - Eric W. Weisstein, Jan 12 2021
a(n) also agrees with the number of optimal fundamentally distinct radio labelings of the n-dipyramidal graph for n = 5 up to at least n = 9 (and likely all larger n). - Eric W. Weisstein, Jan 14 2021

			The 5 cycles of length n=6 having this property are {1,3,5,2,4,6}, {1,3,5,2,6,4}, {1,3,6,4,2,5}, {1,4,2,5,3,6}, {1,4,2,6,3,5}.

Andrew Woods, Table of n, a(n) for n = 1..100 (terms up to a(24) from _Hiroaki Yamanouchi_, Aug 28 2014)
F. C. Holroyd and W. J. G. Wingate, Cycles in the complement of a tree or other graph, Discrete Math., 55 (1985), 267-282.
S. Sykora, On Neighbor-Property Cycles, Stan's Library, Volume V, 2014.
Eric Weisstein's World of Mathematics, Dipyramidal Graph
Eric Weisstein's World of Mathematics, Hamiltonian Cycle
Eric Weisstein's World of Mathematics, Path Complement Graph
Eric Weisstein's World of Mathematics, Radio Labeling
Eric Weisstein's World of Mathematics, Wheel Graph

Cf. A242519, A242520, A242521, A242523, A242524, A242525, A242526, A242527, A242528, A242529, A242530, A242531, A242532, A242533, A242534.

Cf. A006184.

a[n_ /; n < 5] = 0; a[5] = 1; a[6] = 5; a[n_] := a[n] = n a[n - 1] - (n - 5) a[n - 2] - (n - 4) a[n - 3] + (n - 4) a[n - 4]; Array[a, 24] (* Jean-François Alcover, Oct 07 2017 *)
Join[{0, 0}, RecurrenceTable[{a[n] == n a[n - 1] - (n - 5) a[n - 2] - (n - 4) a[n - 3] + (n - 4) a[n - 4], a[3] == a[4] == 0, a[5] == 1, a[6] == 5}, a, {n, 20}]] (* Eric W. Weisstein, Apr 12 2018 *)

a(n) = A002493(n)/(2*n), n>1. - Andrew Woods, Dec 08 2014
a(n) = Sum_{k=1..n} (-1)^(n-k)*k!*A102413(n,k) / (2*n), n>2. - Andrew Woods after Vladeta Jovovic, Dec 08 2014
a(n) = (n-1)!/2 + sum_{i=1..n-1} ((-1)^i * (n-i-1)! * sum_{j=0..i-1} (2^j * C(i-1,j) * C(n-i,j+1))), for n>=5. - Andrew Woods, Jan 08 2015
a(n) = n a(n-1) - (n-5) a(n-2) - (n-4) a(n-3) + (n-4) a(n-4), for n>6. - Jean-François Alcover, Oct 07 2017

A137774 Number of ways to place n nonattacking empresses on an n X n board.

Original entry on oeis.org

1, 2, 2, 8, 20, 94, 438, 2766, 19480, 163058, 1546726, 16598282, 197708058, 2586423174, 36769177348, 563504645310, 9248221393974, 161670971937362, 2996936692836754, 58689061747521430, 1210222434323163704, 26204614054454840842, 594313769819021397534, 14086979362268860896282

Offset: 1

Vaclav Kotesovec, Jan 27 2011

An empress moves like a rook and a knight.

Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, Separators - a new statistic for permutations, arXiv:1905.12364 [math.CO], 2019.
Eli Bagno, Estrella Eisenberg, Shulamit Reches, and Moriah Sigron, On the Sparseness of the Downsets of Permutations via Their Number of Separators, Enumerative Combinatorics and Applications (2021) Vol. 1, No. 3, Article #S2R21.
Vaclav Kotesovec, Non-attacking chess pieces, 6ed, 2013, p.685 and 636.
W. Schubert, N-Queens page

Cf. A201513, A000170, A002465, A201540.
Cf. A185085, A051223, A244284, A201511, A201861, A137774, A245011.
Cf. A218244, A002464, A110128, A117574, A089222, A002493.

Asymptotics (Vaclav Kotesovec, Jan 26 2011): a(n)/n! -> 1/e^4.
General asymptotic formulas for number of ways to place n nonattacking pieces rook + leaper[r,s] on an n X n board:
a(n)/n! -> 1/e^2 for 0a(n)/n! -> 1/e^4 for 0

Terms a(16)-a(17) from Vaclav Kotesovec, Feb 06 2011
Terms a(18)-a(19) from Wolfram Schubert, Jul 24 2011
Terms a(20)-a(24) (computed by Wolfram Schubert), Vaclav Kotesovec, Aug 25 2012

A338838 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] where adjacent values cannot be consecutive modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, 0, 1, 4, 4, 0, 0, 1, 5, 10, 10, 10, 10, 1, 6, 18, 36, 60, 84, 60, 1, 7, 28, 84, 210, 434, 630, 462, 1, 8, 40, 160, 544, 1552, 3440, 5168, 3920, 1, 9, 54, 270, 1170, 4338, 13158, 30366, 47178, 36954, 1, 10, 70, 420, 2220, 10220, 39780, 125220, 298060, 476220, 382740

Offset: 0

Xiangyu Chen, Nov 11 2020

In a convex n-gon, the number of paths using k-1 diagonals and k non-repeated vertices.

			n\k  0    1    2    3    4    5    6    7    8
0    1
1    1    1
2    1    2    0
3    1    3    0    0
4    1    4    4    0    0
5    1    5    10   10   10   10
6    1    6    18   36   60   84   60
7    1    7    28   84   210  434  630  462
8    1    8    40   160  544  1552 3440 5168 3920

Right diagonal is A002493.

Cf. A011973, A034807, A338526, A338849, A000166.

isokd(d, n) = my(x=abs(d)); (x==1) || (x==(n-1));
isok(s, p, n) = {my(w = vector(#s, k, s[p[k]])); for (i=1, #s-1, if (isokd(w[i+1] - w[i], n) == 1, return (0))); return (1);}
T(n, k) = {my(nb = 0); forsubset([n, k], s, for(i=1, k!, if (isok(s, numtoperm(k, i), n), nb++););); nb;} \\ Michel Marcus, Nov 21 2020

T(n,k) = n*(A338526(n-1,k-1)-S(n-1,k-1)) for k>0 except T(2,2)=0, T(n,0)=1, where S(n,k) = 2*A338526(n-1,k-1)-S(n-1,k-1) for k>0, S(n,0)=0.

A384921 Number of permutations [p_1, p_2, ..., p_n], for n >= 1, with |p_{i+1} - p_i| >= 2, for i = 1..n-1, and |p_n - p_1| = 0 or 1.

Original entry on oeis.org

1, 0, 0, 2, 4, 30, 184, 1322, 10668, 96566, 969280, 10690146, 128527348, 1673257262, 23451539784, 352079626010, 5637207651004, 95886993887142, 1726775043225808, 32821564079286866, 656647922936247300, 13793480376190668446

Offset: 1

Wolfdieter Lang, Jun 17 2025

This sequence gives the number of the so-called king permutations, for n >= 1, counted in A002464, that satisfy the additional restriction |p_n - p_1| = 0 or 1.

			n=1: The permutation is [1].
n=4: The two king permutations are [2, 4, 1, 3] and its reversal [3, 1, 4, 2].
n=5: The four permutations are [2,,4, 1, 5, 3], [3, 1, 5, 2, 4] and their reversals [3, 5, 1, 4, 2], [4, 2, 5, 1, 3]. See III of the Abramson and Moser link, p. 1254.
n=6: The 30 permutations are (in short cut version): 146352, 153642, 246153, 251463, 264153, 315264, 351624, 352614, 361524, 362514, 413625, 426135, 426315, 524136, 531426, and their reversals.

M. Abramson and W. O. J. Moser, Permutations without rising or falling w-sequences, Ann. Math. Stat., 38 (1967), 1245-1254, p. 1254, III.

Cf. A002464, A002493.

a(n) = A002464(n) - A002493(n), for n >= 2, but A002493(1) = 1, not 0, as it is here, if instead of A002493 the definition |p_{i+1} - p_i| >= 2, for i = 1..n, for n >= 1, and p_{n+1} = p_1 is used; hence a(1) = 1, not 0.

a(n) = 2*Sum_{k=0..floor((n-2)/2)} A002464(n - (2*k+1)), for n >= 3, and a(1) = 1, a(2) = 0. (Compare this with the formula given by Vladeta Jovovic in A002493, Nov 24 2007.)

Showing 1-5 of 5 results.

cp's OEIS Frontend

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.

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A242522 Number of cyclic arrangements of S={1,2,...,n} such that the difference between any two neighbors is at least 2.

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A137774 Number of ways to place n nonattacking empresses on an n X n board.

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A338838 Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] where adjacent values cannot be consecutive modulo n.

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A384921 Number of permutations [p_1, p_2, ..., p_n], for n >= 1, with |p_{i+1} - p_i| >= 2, for i = 1..n-1, and |p_n - p_1| = 0 or 1.

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