A000387 -id:A000387 - cp's OEIS frontend

A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 4, 1, 0, 1, 1, 2, 5, 8, 1, 0, 1, 1, 2, 5, 14, 16, 1, 0, 1, 1, 2, 5, 15, 41, 32, 1, 0, 1, 1, 2, 5, 15, 51, 122, 64, 1, 0, 1, 1, 2, 5, 15, 52, 187, 365, 128, 1, 0, 1, 1, 2, 5, 15, 52, 202, 715, 1094, 256, 1, 0

Offset: 0

Peter Luschny, Nov 05 2018

Arndt and Sloane (see the link and A278984) identify the sequence to give "the number of words of length n over an alphabet of size b that are in standard order" and provide the formula Sum_{j = 1..b} Stirling_2(n, j) assuming b >= 1 and j >= 1. Compared to the array as defined here this misses the first row and the first column of our array.
The method used here is the special case of a general method described in A320956 applied to the function exp. For applications to other functions see the cross references.
A(k,n) is the number of color patterns (set partitions) for an oriented row of length n using up to k colors (subsets). Two color patterns are equivalent if the colors are permuted. For A(3,4) = 14, the six achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA; the eight chiral patterns are the four chiral pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - Robert A. Russell, Nov 10 2018

			Array starts:
n\k   0  1  2  3   4   5    6    7     8      9  ...
----------------------------------------------------
[0]   1, 0, 0, 0,  0,  0,   0,   0,    0,     0, ...  A000007
[1]   1, 1, 1, 1,  1,  1,   1,   1,    1,     1, ...  A000012
[2]   1, 1, 2, 4,  8, 16,  32,  64,  128,   256, ...  A011782
[3]   1, 1, 2, 5, 14, 41, 122, 365, 1094,  3281, ...  A124302
[4]   1, 1, 2, 5, 15, 51, 187, 715, 2795, 11051, ...  A124303
[5]   1, 1, 2, 5, 15, 52, 202, 855, 3845, 18002, ...  A056272
[6]   1, 1, 2, 5, 15, 52, 203, 876, 4111, 20648, ...  A056273, ?A284727
[7]   1, 1, 2, 5, 15, 52, 203, 877, 4139, 21110, ...
[8]   1, 1, 2, 5, 15, 52, 203, 877, 4140, 21146, ...
[9]   1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, ...
----------------------------------------------------
Seen as a triangle given by the descending antidiagonals:
[0]             1
[1]            0, 1
[2]          0, 1, 1
[3]        0, 1, 1, 1
[4]       0, 1, 2, 1, 1
[5]     0, 1, 4, 2, 1, 1
[6]    0, 1, 8, 5, 2, 1, 1
[7]  0, 1, 16, 14, 5, 2, 1, 1

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Cf. A008290, A000110, A137650, A211561, A000240, A000387, A000449, A103816/A053556.
Cf. A124302, A124303, A056272, A056273, A284727, A278984.
Antidiagonal sums (and row sums of the triangle): A320964.
Cf. this sequence (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), A320959 (arctanh).
Cf. A320750 (unoriented), A320751 (chiral), A305749 (achiral).

A := (n, k) -> if k = 0 then 1 else add(A008290(n, n-j)*(n-j)^k, j=0..n-1)/n! fi:
seq(lprint(seq(A(n, k), k=0..9)), n=0..9); # Prints the array row-wise.
seq(seq(A(n-k, k), k=0..n), n=0..11); # Gives the array as listed.

T[n_, 0] := 1; T[n_, k_] := Sum[(Subfactorial[j]/Factorial[j])((n - j)^k/(n - j)!), {j, 0, n - 1}]; Table[T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten
Table[Sum[StirlingS2[k, j], {j, 0, n-k}], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert A. Russell, Nov 10 2018 *)

A(n, k) = (1/n!)*Sum_{j=0..n-1} A008290(n, n-j)*(n-j)^k if k > 0.
If one drops the special case A(n, 0) = 1 from the definition then column 0 becomes Sum_{k=0..n} (-1)^k/k! = A103816(n)/A053556(n).
Row n is given for k >= 1 by a_n(k), where
a_0(k) = 0^k/0!.
a_1(k) = 1^k/1!.
a_2(k) = (2^k)/2!.
a_3(k) = (3^k + 3)/3!.
a_4(k) = (6*2^k + 4^k + 8)/4!.
a_5(k) = (20*2^k + 10*3^k + 5^k + 45)/5!.
a_6(k) = (135*2^k + 40*3^k + 15*4^k + 6^k + 264)/6!.
a_7(k) = (924*2^k + 315*3^k + 70*4^k + 21*5^k + 7^k + 1855)/7!.
a_8(k) = (7420*2^k + 2464*3^k + 630*4^k + 112*5^k + 28*6^k + 8^k + 14832)/8!.
Note that the coefficients of the generating functions a_n are the recontres numbers A000240, A000387, A000449, ...
Rewriting the formulas with exponential generating functions for the rows we have egf(n) = Sum_{k=0..n} !k*binomial(n,k)*exp(x*(n-k)) and A(n, k) = (k!/n!)*[x^k] egf(n). In this formulation no special rule for the case k = 0 is needed.
The rows converge to the Bell numbers. Convergence here means that for every fixed k the terms in column k differ from A000110(k) only for finitely many indices.
A(n, n) are the Bell numbers A000110(n) for n >= 0.
Let S(n, k) = Bell(n+k+1) - A(n, k+n+1) for n >= 0 and k >= 0, then the square array S(n, k) read by descending antidiagonals equals provable the triangle A137650 and equals empirical the transpose of the array A211561.

A145225 Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 6, 0, 6, 0, 0, 20, 30, 0, 10, 0, 0, 135, 120, 90, 0, 15, 0, 0, 924, 945, 420, 210, 0, 21, 0, 0, 7420, 7392, 3780, 1120, 420, 0, 28, 0, 0, 66744, 66780, 33264, 11340, 2520, 756, 0, 36, 0, 0

Offset: 0

Abdullahi Umar, Oct 10 2008

			Triangle starts:
   0;
   0,  0;
   1,  0, 0;
   0,  3, 0,  0;
   6,  0, 6,  0, 0;
  20, 30, 0, 10, 0, 0;
  ...

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of triangle, flattened).
Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Row sums are A001710 for n > 1.
Columns k=0..2 are A000387, A145222, A145223.
Cf. A000387, A008290, A055137, A145224.

A145225 := proc(n,k)
    binomial(n,k)*A000387(n-k) ; # re-use code of A000387
end proc:
seq(seq(A145225(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 06 2023

A145225[n_, k_] := Binomial[n, k]*Binomial[n - k, 2]*Subfactorial[n - k - 2];
Table[A145225[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Jan 31 2025 *)

T(n,k) = C(n,k) * A000387(n-k).
E.g.f.: x^(k+2) * exp(-x) / (2*k!*(1-x)).
T(n,k) + A145224(n,k) = A008290(n,k). - R. J. Mathar, Jul 06 2023
T(n,k) = (A008290(n,k) - A055137(n,k)) / 2. - Julian Hatfield Iacoponi, Aug 08 2024

A065087 a(n) = A000166(n)*binomial(n+1,2).

Original entry on oeis.org

0, 0, 3, 12, 90, 660, 5565, 51912, 533988, 6007320, 73422855, 969181620, 13744757598, 208462156812, 3367465610145, 57727981888080, 1046800738237320, 20020064118788592, 402756584036805963, 8502638996332570140, 187953072550509445410, 4341715975916768188740

Offset: 0

N. J. A. Sloane, Nov 10 2001

nonn

a(n) is also the number of permutations of [2n-1] having n-1 isolated fixed points (i.e. adjacent entries are not fixed points). Example: a(2)=3 because we have 132, 213, and 321. - Emeric Deutsch, Apr 18 2009

Seiichi Manyama, Table of n, a(n) for n = 0..400

Equals 3 * A000313(n+2).

Cf. A000166, A000240, A000387, A305730.

a[n_] := Subfactorial[n]*Binomial[n + 1, 2];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 18 2024 *)

a(n) = (n/2)*A000240(n+1). - Zerinvary Lajos, Dec 18 2007, corrected Jul 09 2012
a(n) = n * (n+1) * (a(n-1)/(n-1) + (-1)^n/2) for n > 1 - Seiichi Manyama, Jun 24 2018
E.g.f.: exp(-x)*x^2*(3 - 2*x + x^2)/(2*(1 - x)^3). - Ilya Gutkovskiy, Jun 25 2018

A145222 a(n) is the number of odd permutations (of an n-set) with exactly 1 fixed point.

Original entry on oeis.org

0, 0, 3, 0, 30, 120, 945, 7392, 66780, 667440, 7342335, 88107360, 1145396538, 16035550440, 240533257965, 3848532125760, 65425046139960, 1177650830516832, 22375365779822715, 447507315596450880, 9397653627525472470, 206748379805560389720, 4755212735527888968873

Offset: 1

Abdullahi Umar, Oct 09 2008

nonn

			a(3) = 3 because there are exactly 3 odd permutations (of a 3-set) having 1 fixed point, namely: (12), (13), (23).

Paolo Xausa, Table of n, a(n) for n = 1..400
Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Cf. A000387 (odd permutations with no fixed points), A145219 (even permutations with exactly 1 fixed point), A145223 (odd permutations with exactly 2 fixed points).

A145222[n_] := n*Subfactorial[n - 3]*Binomial[n - 1, 2]; Array[A145222, 25] (* Paolo Xausa, Jan 31 2025 *)

x = 'x + O('x^30); Vec(serlaplace(((x^3)*exp(-x))/(2*(1-x)))) \\ Michel Marcus, Apr 04 2016

a(n) = A145225(n,1) = n*A000387(n-1), (n > 0).
E.g.f.: x^3*exp(-x)/(2*(1-x)).
D-finite with recurrence (-n+3)*a(n) +n*(n-4)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

More terms from Alois P. Heinz, Apr 04 2016

A180189 Number of permutations of [n] having exactly 1 circular succession. A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

Original entry on oeis.org

0, 2, 0, 12, 40, 270, 1848, 14840, 133488, 1334970, 14684560, 176214852, 2290792920, 32071101062, 481066515720, 7697064251760, 130850092279648, 2355301661033970, 44750731559645088, 895014631192902140, 18795307255050944520, 413496759611120779902

Offset: 1

Emeric Deutsch, Sep 06 2010

nonn

For example, p=(4,1,2,5,3) has 2 circular successions: (1,2) and (3,4).

			a(4)=12 because we have 1*243, 142*3, 13*42, 31*24, 3142*, 431*2, 213*4, 4213*, 2*314, 2431*, 42*31, and 3*421 (the circular succession is marked *).

S. M. Tanny, Permutations and successions, J. Combinatorial Theory, Series A, 21 (1976), 196-202.

Cf. A000166, A000387, A180188.

d[0] := 1: for n to 51 do d[n] := n*d[n-1]+(-1)^n end do: seq(n*(n-1)*d[n-2], n = 1 .. 22);

a(n) = n*(n-1)*d(n-2), where d(j)=A000166(j) are the derangement numbers.
a(n) = A180188(n,1).
E.g.f.: x^2 * exp(-x) / (1 - x). - Ilya Gutkovskiy, Oct 11 2021
a(n) = 2 * A000387(n). - Alois P. Heinz, Oct 11 2021
D-finite with recurrence (-n+2)*a(n) +n*(n-3)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 26 2022

A060836 Number of permutations of n letters where exactly 5 change position.

Original entry on oeis.org

0, 0, 0, 0, 44, 264, 924, 2464, 5544, 11088, 20328, 34848, 56628, 88088, 132132, 192192, 272272, 376992, 511632, 682176, 895356, 1158696, 1480556, 1870176, 2337720, 2894320, 3552120, 4324320, 5225220, 6270264, 7476084, 8860544, 10442784, 12243264, 14283808, 16587648

Offset: 1

Robert Goodhand (rgoodhand(AT)hotmail.com), May 12 2001

			a(8) = a(7) * 8/(8-5) = 924 * 8/3 = 2464.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

For changing 0, 1, 2, 3, 4, 5, n-4, n elements see A000012, A000004, A000217 (offset), A007290, A060008, A060836, A000475, A000166. Also see A000332, A008290.
Rencontre sequences are A000166 A000240 A000387 A000449 and A000475.
A diagonal of A008291.

a(n) = { 44*binomial(n, 5) } \\ Harry J. Smith, Jul 19 2009

a(n) = 44*binomial(n, 5).
a(n) = a(n-1)*n/(n-5).
G.f.: 44*x^5/(1 - x)^6. - Colin Barker, Apr 22 2012

A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

Original entry on oeis.org

0, 0, 6, 0, 90, 420, 3780, 33264, 333900, 3670920, 44054010, 572697840, 8017775766, 120266628300, 1924266063720, 32712523068960, 588825415259640, 11187682889909904, 223753657798227150, 4698826813762734240, 103374189902780197170, 2377606367763944481780

Offset: 2

Abdullahi Umar, Oct 09 2008

nonn

			a(4) = 6 because there are exactly 6 odd permutations (of a 4-set) having 2 fixed points, namely: (12), (13), (14), (23), (24), (34).

Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.

Cf. A000387 (odd permutations with no fixed points), A145222 (odd permutations with exactly 1 fixed point), A145220 (even permutations with exactly 2 fixed points).

egf:= x^4 * exp(-x)/(4*(1-x));
a:= n-> n! * coeff(series(egf, x, n+1), x, n):
seq(a(n), n=2..30);  # Alois P. Heinz, Feb 01 2011

A000387[n_] := Subfactorial[n-2] Binomial[n, 2];
a[n_] := (n(n-1)/2) A000387[n-2];
Table[a[n], {n, 2, 30}] (* Jean-François Alcover, Jan 30 2025 *)

x = 'x + O('x^30); Vec(serlaplace(((x^4)*exp(-x))/(4*(1-x)))) \\ Michel Marcus, Apr 04 2016

a(n) = A145225(n,2) = (n*(n-1)/2) * A000387(n-2), (n > 1).
E.g.f.: x^4*exp(-x)/(4*(1-x)).
D-finite with recurrence +(-n+6)*a(n) +(n-2)*(n-7)*a(n-1) +(n-2)*(n-3)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

More terms from Alois P. Heinz, Feb 01 2011

A145880 Triangle read by rows: T(n,k) is the number of odd permutations of {1,2,...,n} with no fixed points and having k excedances (n>=1; k>=1).

Original entry on oeis.org

0, 1, 0, 0, 1, 4, 1, 0, 10, 10, 0, 1, 26, 81, 26, 1, 0, 56, 406, 406, 56, 0, 1, 120, 1681, 3816, 1681, 120, 1, 0, 246, 6210, 26916, 26916, 6210, 246, 0, 1, 502, 21433, 160054, 303505, 160054, 21433, 502, 1, 0, 1012, 70774, 852346, 2747008, 2747008, 852346, 70774

Offset: 1

Emeric Deutsch, Nov 06 2008

Row n has n-1 entries (n>=2).
Sum of entries in row n = A000387(n).
Sum_{k=1..n-1} k*T(n,k) = A145886(n) (n>=2).

			T(4,2)=4 because the odd derangements of {1,2,3,4} with 2 excedances are 3142, 4312, 2413 and 3421.
Triangle starts:
  0;
  1;
  0,  0;
  1,  4,  1;
  0, 10, 10,  0;
  1, 26, 81, 26,  1;

R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188.

Cf. A000387, A145886, A145881, A145887.

G:=((1-t)*exp(-t*z)/(1-t*exp((1-t)*z))+(t*exp(-z)-exp(-t*z))/(1-t))*1/2: Gser:=simplify(series(G,z=0,15)): for n to 11 do P[n]:=sort(expand(factorial(n)*coeff(Gser,z,n))) end do: 0; for n to 11 do seq(coeff(P[n],t,j),j=1..n-1) end do; # yields sequence in triangular form

E.g.f.: ((1-t)*exp(-tz)/(1-t*exp((1-t)z)) + (t*exp(-z)-exp(-tz))/(1-t))/2.

Formula corrected by N. J. A. Sloane, Jul 20 2017 at the request of the author.

A161937 The number of indirect isometries that are derangements of the (n-1)-dimensional facets of an n-cube.

Original entry on oeis.org

1, 2, 15, 116, 1165, 13974, 195643, 3130280, 56345049, 1126900970, 24791821351, 595003712412, 15470096522725, 433162702636286, 12994881079088595, 415836194530835024, 14138430614048390833, 508983502105742069970, 19341373080018198658879, 773654923200727946355140

Offset: 1

Elizabeth McMahon, Gary Gordon (mcmahone(AT)lafayette.edu), Jun 29 2009

a(n) plays the same role as A000387 plays for the derangement numbers A000166.

			For a square, the 2 diagonal reflections are indirect edge derangements. For a 3-cube, the 15 rotary reflections are indirect face derangements.

G. Gordon and E. McMahon, Moving faces to other places: facet derangements, arXiv:0906.4253 [math.CO], 2009.
Gary Gordon and Elizabeth McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.

Cf. A000354, A000387, A161936.

a := n -> (-1)^(n+1)*n*hypergeom([1,1-n],[],2):
seq(simplify(a(n)), n=1..20); # Peter Luschny, May 09 2017

a[n_] := (-1)^(n + 1)*n*HypergeometricPFQ[{1, 1 - n}, {}, 2];
Array[a, 20] (* Jean-François Alcover, Jul 14 2018, after Peter Luschny *)

a(n) = (b(n) + (-1)^(n+1))/2, where b(n) is sequence A000354, i.e., the number of (n-1)-dimensional facet derangements of an n-cube.
From Peter Luschny, May 09 2017: (Start)
a(n) = (-1)^(n+1)*n*hypergeom([1, 1-n], [], 2).
a(n) = (2^n*Gamma(n+1,-1/2)/exp(1/2)-(-1)^n)/2. (End)
E.g.f.: x*exp(-x) / (1 - 2*x). - Ilya Gutkovskiy, May 23 2020

More terms from Peter Luschny, May 09 2017

A162974 Triangle read by rows: T(n,k) is the number of derangements of {1,2,...,n} having k cycles of length 2 (0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 6, 0, 3, 24, 20, 0, 160, 90, 0, 15, 1140, 504, 210, 0, 8988, 4480, 1260, 0, 105, 80864, 41040, 9072, 2520, 0, 809856, 404460, 100800, 18900, 0, 945, 8907480, 4447520, 1128600, 166320, 34650, 0, 106877320, 53450496, 13347180, 2217600

Offset: 0

Emeric Deutsch, Jul 22 2009

Row n has 1 + floor(n/2) entries.
Sum of entries in row n = A000166(n) (the derangement numbers).
T(n,0) = A038205(n).
Sum_{k>=0} k*T(n,k) = A000387(n).

			T(4,2)=3 because we have (12)(34), (13)(24), and (14)(23).
Triangle starts:
    1;
    0;
    0,  1;
    2,  0;
    6,  0,  3;
   24, 20,  0;
  160, 90,  0, 15;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000166, A000387, A038205, A114320.
T(2n,n) gives A001147.
T(2n+3,n) gives A000906(n) = 2*A000457(n).

G := exp((1/2)*z*(t*z-z-2))/(1-z): Gser := simplify(series(G, z = 0, 16)): for n from 0 to 13 do P[n] := sort(expand(factorial(n)*coeff(Gser, z, n))) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do;
# second Maple program:
b:= proc(n) option remember; expand(`if`(n=0, 1, add((j-1)!*
      `if`(j=2, x, 1)*b(n-j)*binomial(n-1, j-1), j=2..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n/2))(b(n)):
seq(T(n), n=0..14);  # Alois P. Heinz, Jan 27 2022

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[(j - 1)!*If[j == 2, x, 1]*b[n - j]*Binomial[n - 1, j - 1], {j, 2, n}]]];
T[n_] := With[{p = b[n]}, Table[Coefficient[p, x, i], {i, 0, n/2}]];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Sep 17 2024, after Alois P. Heinz *)

E.g.f.: G(t,z) = exp(z(tz-z-2)/2)/(1-z).

cp's OEIS Frontend

A320955 Square array read by ascending antidiagonals: A(n, k) (n >= 0, k >= 0) = Sum_{j=0..n-1} (!j/j!)*((n - j)^k/(n - j)!) if k > 0 and 1 if k = 0. Here !n denotes the subfactorial of n.

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A145225 Triangle read by rows: T(n,k) is the number of odd permutations (of an n-set) with exactly k fixed points.

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A065087 a(n) = A000166(n)*binomial(n+1,2).

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A145222 a(n) is the number of odd permutations (of an n-set) with exactly 1 fixed point.

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A180189 Number of permutations of [n] having exactly 1 circular succession. A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.

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A060836 Number of permutations of n letters where exactly 5 change position.

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A145223 a(n) is the number of odd permutations (of an n-set) with exactly 2 fixed points.

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PARI

Formula

Extensions