A000349 -id:A000349 - 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

A086852 Number of permutations of length n with exactly 1 rising or falling succession.

Original entry on oeis.org

0, 0, 2, 4, 10, 40, 230, 1580, 12434, 110320, 1090270, 11876980, 141373610, 1825321016, 25405388150, 379158271420, 6039817462210, 102278890975360, 1834691141852174, 34752142215026180, 693126840194499290, 14519428780464454600, 318705819455462421670

Offset: 0

N. J. A. Sloane, Aug 19 2003

Permutations of 12...n such that exactly one of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

For the number of such permutations without (n-1)n or n(n-1) see A383857(n), for n >= 1. - Wolfdieter Lang, May 22 2025

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Sergey Kitaev, Jeffrey Remmel, (a,b)-rectangle patterns in permutations and words, arXiv:1304.4286 [math.CO], 2013.
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.

Cf. A002464, A086853, A086854, A000349, A001267, A383857.
Twice A000130. A diagonal of A001100.
Cf. A104698, A384494.

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((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)))
    end:
a:= n-> coeff(S(n), t, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 21 2012

S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(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_] := Coefficient[S[n], t, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

Coefficient of t^1 in S[n](t) defined in A002464.
(3-n)*a(n) +(n+1)*(n-3)*a(n-1) -(n^2-4*n+5)*a(n-2) -(n-1)*(n-5)*a(n-3) +(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 06 2013
a(n) ~ 2*sqrt(2*Pi)*n!/exp(2) = 0.678470495... * n!. - Vaclav Kotesovec, Aug 10 2013
From Wolfdieter Lang, May 31 2025: (Start)
a(n) = Sum_{i=1..n-1} (-1)^(i-1)*i*(n-i)!*Sum_{j=1..i} 2^j*binomial(i-1, j-1)*binomial(n-i, j), for n >= 0. See the D. P. Robbins link, p. 123, eq. (7), A(n, 1).
a(n+2) = Sum_{k=0..n} 2*R(n, k)*B(n, k), with B(n, k) = A384494(n, k) = (-1)^k*(k+1)*(n+1-k)!, and R(n, k) = A104698(n, k), for n >= 0. This equals (2*MR*MB^t)_{n,n}, with the (infinite) square matrices with vanishing upper diagonals corresponding to R and B, and t indicates transposition. (End)

A010028 Triangle read by rows: T(n,k) is one-half the number of permutations of length n with exactly n-k rising or falling successions, for n >= 1, 1 <= k <= n. T(1,1) = 1 by convention.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 5, 5, 1, 1, 8, 24, 20, 7, 1, 11, 60, 128, 115, 45, 1, 14, 113, 444, 835, 790, 323, 1, 17, 183, 1099, 3599, 6423, 6217, 2621, 1, 20, 270, 2224, 11060, 32484, 56410, 55160, 23811, 1, 23, 374, 3950, 27152, 118484, 325322, 554306, 545135, 239653

Offset: 1

N. J. A. Sloane

(1/2) times number of permutations of 12...n such that exactly n-k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

			Triangle T(n,k) begins:
  1;
  1,  0;
  1,  2,   0;
  1,  5,   5,    1;
  1,  8,  24,   20,    7;
  1, 11,  60,  128,  115,   45;
  1, 14, 113,  444,  835,  790,  323;
  1, 17, 183, 1099, 3599, 6423, 6217, 2621;
  ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz, Rows n = 1..141, flattened
J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

Diagonals give A001266 (and A002464), A000130, A000349, A001267, A001268.
Triangle in A086856 transposed. Cf. A001100.
Row sums give A001710.

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((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)))
    end:
T:= (n, k)-> ceil(coeff(S(n), t, n-k)/2):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Dec 21 2012

S[n_] := S[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(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]]]; T[n_, k_] := Ceiling[Coefficient[S[n], t, n-k]/2]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

For n>1, coefficient of t^(n-k) in S[n](t) defined in A002464, divided by 2.

A086853 Number of permutations of length n with exactly 2 rising or falling successions.

Original entry on oeis.org

0, 0, 0, 2, 10, 48, 256, 1670, 12846, 112820, 1108612, 12032154, 142852450, 1840969784, 25587270600, 381460235918, 6071318154166, 102742200205980, 1841978156709676, 34874169034136930, 695294184953602602, 14560120360421802464, 319510983674891800240

Offset: 0

N. J. A. Sloane, Aug 19 2003

nonn

Permutations of 12...n such that exactly 2 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz, Table of n, a(n) for n = 0..200
J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

Cf. A002464, A000130, A000349, A001267, A086852, A086854. A diagonal of A001100.

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((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)))
    end:
a:= n-> ceil(coeff(S(n), t, 2)):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 11 2013

s[n_] := s[n] = If[n<4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(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]]]; t[n_, k_] := Ceiling[Coefficient[s[n], t, k]]; a[n_] := t[n, 2]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)

Coefficient of t^2 in S[n](t) defined in A002464.
Conjecture: (-514*n+2465)*a(n) +2*(257*n^2-955*n-1085)*a(n-1) +(-555*n^2+2483*n-1670)*a(n-2) +16*(-17*n^2+73*n-75)*a(n-3) +(354*n^2+528*n-2299)*a(n-4) +2*(-121*n^2+1045*n-1401)*a(n-5) +3*(67*n-115)*(n-4)*a(n-6)=0. - R. J. Mathar, Jun 06 2013
shorter recurrence: (n-3)*(n-2)*(n-4)^3*a(n) = (n-3)*(n^4-9*n^3+23*n^2-4*n-29)*(n-4)*a(n-1) - (n-1)*(n^4-12*n^3+57*n^2-125*n+104)*(n-4)*a(n-2) - (n-2)*(n-1)*(n^4-15*n^3+83*n^2-198*n+169)*a(n-3) + (n-3)^3*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Aug 10 2013
a(n) ~ 2*exp(-2) * n!. - Vaclav Kotesovec, Aug 10 2013

A086854 Number of permutations of length n with exactly 3 rising or falling successions.

Original entry on oeis.org

0, 0, 0, 0, 2, 16, 120, 888, 7198, 64968, 650644, 7165200, 86059242, 1119549472, 15682257872, 235336043976, 3766695159030, 64052134910168, 1153211148654348, 21915344800505888, 438380075974889154, 9207290871553008240, 202585136417883766472, 4659950328485470292632

Offset: 0

N. J. A. Sloane, Aug 19 2003

nonn

Permutations of 12...n such that exactly 3 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
J. Riordan, A recurrence for permutations without rising or falling successions. Ann. Math. Statist. 36 (1965), 708-710.

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A002464, A000130, A000349, A001267, A086852, A086853. A diagonal of A001100.

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((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)))
    end:
a:= n-> coeff(S(n), t, 3):
seq(a(n), n=0..25);  # Alois P. Heinz, Jan 11 2013

S[n_] := S[n] = If[n < 4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(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_] := Coefficient[S[n], t, 3]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 09 2014, after Alois P. Heinz *)

Coefficient of t^3 in S[n](t) defined in A002464.
Recurrence (for n>4): (n-4)*(2*n^6 - 52*n^5 + 557*n^4 - 3136*n^3 + 9740*n^2 - 15727*n + 10242)*a(n) = (n-4)*(2*n^7 - 50*n^6 + 511*n^5 - 2693*n^4 + 7450*n^3 - 9041*n^2 - 157*n + 6666)*a(n-1) - (2*n^8 - 58*n^7 + 735*n^6 - 5289*n^5 + 23430*n^4 - 64575*n^3 + 106105*n^2 - 92312*n + 30900)*a(n-2) - (2*n^7 - 54*n^6 + 615*n^5 - 3795*n^4 + 13554*n^3 - 27681*n^2 + 29473*n - 12330)*(n-2)*a(n-3) + (2*n^6 - 40*n^5 + 327*n^4 - 1388*n^3 + 3184*n^2 - 3675*n + 1626)*(n-2)^2*a(n-4). - Vaclav Kotesovec, Aug 11 2013
a(n) ~ 4/3*exp(-2) * n! = n! * 0.45231366335478... - Vaclav Kotesovec, Aug 11 2013

A086856 Triangle read by rows: T(n,k) = one-half number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1. T(1,0) = 1 by convention.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 1, 5, 5, 1, 7, 20, 24, 8, 1, 45, 115, 128, 60, 11, 1, 323, 790, 835, 444, 113, 14, 1, 2621, 6217, 6423, 3599, 1099, 183, 17, 1, 23811, 55160, 56410, 32484, 11060, 2224, 270, 20, 1, 239653, 545135, 554306, 325322, 118484, 27152, 3950, 374, 23, 1, 2648395

Offset: 1

N. J. A. Sloane, Aug 19 2003

(1/2) times number of permutations of 1, 2, ..., n such that exactly k of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

			Triangle T(n,k) begins:
    1;
    0,   1;
    0,   2,   1;
    1,   5,   5,   1;
    7,  20,  24,   8,   1;
   45, 115, 128,  60,  11,  1;
  323, 790, 835, 444, 113, 14, 1;
  ...

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

Alois P. Heinz, Rows n = 1..141, flattened
J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

Columns give A001266 (see also A002464), A000130, A000349, A001267, A001268.
Triangle in A001100 divided by 2 (except for T(1, 0)). A010028 transposed.
Row sums give A001710.

S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
       [n+1], expand((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)))
    end:
T:= (n, k)-> ceil(coeff(S(n), t, k)/2):
seq(seq(T(n, k), k=0..n-1), n=1..10);  # Alois P. Heinz, Jan 11 2013

S[n_] := S[n] = If[n < 4, {1, 1, 2*t, 4*t+2*t^2}[[n+1]], Expand[(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]]]; T[n_, k_] := Ceiling[Coefficient[S[n], t, k]/2]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 11}] // Flatten (* Jean-François Alcover, Jan 14 2014, translated from Alois P. Heinz's Maple code *)

By diagonals (d=1,2,3,...) going to the left with (1,3,6,...) = d(1), these are sequences of the form (k-th term a(k) = d*a(k-1) + k). Example: 1, 7, 38, 194, ... (the 5th diagonal) = A014827, is generated by a(k) = 5*a(k-1) + k. Diagonal 2 = (1, 4, 11, 26, ...) = A000295; Diagonal 3 = (1, 5, 18, ...) = A000340; Diagonal 4 = (1, 6, 27, ...) = A014825.

Triangle A108243 is generated by analogous operations from (..., 3, 2, 1) instead of (1, 2, 3, ...).

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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A086852 Number of permutations of length n with exactly 1 rising or falling succession.

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A010028 Triangle read by rows: T(n,k) is one-half the number of permutations of length n with exactly n-k rising or falling successions, for n >= 1, 1 <= k <= n. T(1,1) = 1 by convention.

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A086853 Number of permutations of length n with exactly 2 rising or falling successions.

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A086854 Number of permutations of length n with exactly 3 rising or falling successions.

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A086856 Triangle read by rows: T(n,k) = one-half number of permutations of length n with exactly k rising or falling successions, for n >= 1, 0 <= k <= n-1. T(1,0) = 1 by convention.

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A086855 Number of permutations of length n with exactly 4 rising or falling successions.

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