A002467 -id:A002467 - cp's OEIS frontend

A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

0, 1, 3, 13, 67, 411, 2921, 23633, 214551, 2160343, 23897269, 288102189, 3760013027, 52816397219, 794536751217, 12744659120521, 217140271564591, 3916221952414383, 74539067188152941, 1493136645424092773, 31400620285465593339, 691708660911435955579

Offset: 1

Emeric Deutsch, Sep 07 2010

nonn

a(n) = A180190(n,1).
a(n+2) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = k! for k = 1, ..., n+1. - Michael Somos, Jan 05 2012
From Jon Perry, Jan 04 2013: (Start)
Number of permutations of {1,...,n-1,n+1} with at least one indexed point p(k)=k with 1<=k<=n. Note that this means p(k)=n+1 is never an indexed point as kFor n>1, a(n) is the number of permutations of [n+1] that have a fixed point and contain 12; for example the a(3)=3 such permutations of {1,2,3,4} are 1234, 1243, and 3124.
(End)
For n > 0: row sums of triangle A116853. - Reinhard Zumkeller, Aug 31 2014

			x^2 + 3*x^3 + 13*x^4 + 67*x^5 + 411*x^6 + 2921*x^7 + 23633*x^8 + ...
a(3) = 3 because we have 123, 312, and 231; the permutations 132, 213, and 321 have no successions.
a(4) = 13 since p(x) = (3*x^2 - 7*x + 6) / 2 interpolates p(1) = 1, p(2) = 2, p(3) = 6, and p(4) = 13. - _Michael Somos_, Jan 05 2012

Alois P. Heinz, Table of n, a(n) for n = 1..200
Uriel Feige, Tighter bounds for online bipartite matching, 2018.

Cf. A000142, A000166, A000255, A002467, A028417, A180190.
Cf. A116853, A276975.
Column k=1 of A306234, A306461, and of A324362(n-1).

a180191 n = if n == 1 then 0 else sum $ a116853_row (n - 1)
-- Reinhard Zumkeller, Aug 31 2014

d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: seq(factorial(n)-d[n]-d[n-1], n = 1 .. 22);

f[n_] := Sum[ -(-1)^k (n - k)! Binomial[n - 1, k], {k, 1, n}]; Array[f, 20] (* Robert G. Wilson v, Oct 16 2010 *)

{a(n) = if( n<2, 0, n--; subst( polinterpolate( vector( n, k, k!)), x, n+1))} /* Michael Somos, Jan 05 2012 */

a(n) = n! - d(n) - d(n-1), where d(j) = A000166(j) are the derangement numbers.
a(n) = n! - A000255(n-1) = A002467(n) - A000166(n-1). - Jon Perry, Jan 05 2013
a(n) = (n-1)! [x^(n-1)] (1-exp(-x))/(1-x)^2. - Alois P. Heinz, Feb 23 2019

A324563 Number T(n,k) of permutations p of [n] such that k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+n*[i=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 1, 7, 15, 0, 1, 1, 11, 31, 76, 0, 1, 1, 18, 60, 185, 455, 0, 1, 1, 29, 113, 435, 1275, 3186, 0, 1, 1, 47, 215, 1001, 3473, 10095, 25487, 0, 1, 1, 76, 406, 2299, 9289, 31315, 90109, 229384, 0, 1, 1, 123, 763, 5320, 24610, 95747, 313227, 895169, 2293839

Offset: 0

Alois P. Heinz, Mar 06 2019

Mirror image of triangle A324564.

Or as array: Number A(n,k) of permutations p of [n+k] such that k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+(n+k)*[i=0, k>=0, read by antidiagonals upwards.

			T(4,1) = A(3,1) = 1: 1234.
T(4,2) = A(2,2) = 1: 4123.
T(4,3) = A(1,3) = 7: 1423, 1432, 3124, 3214, 3412, 4132, 4213.
T(4,4) = A(0,4) = 15: 1243, 1324, 1342, 2134, 2143, 2314, 2341, 2413, 2431, 3142, 3241, 3421, 4231, 4312, 4321.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1,  4;
  0, 1, 1,  7,  15;
  0, 1, 1, 11,  31,   76;
  0, 1, 1, 18,  60,  185,   455;
  0, 1, 1, 29, 113,  435,  1275,   3186;
  0, 1, 1, 47, 215, 1001,  3473,  10095, 25487;
  ...
Square array A(n,k) begins:
  1, 1, 1,  4,  15,   76,   455,   3186, ...
  0, 1, 1,  7,  31,  185,  1275,  10095, ...
  0, 1, 1, 11,  60,  435,  3473,  31315, ...
  0, 1, 1, 18, 113, 1001,  9289,  95747, ...
  0, 1, 1, 29, 215, 2299, 24610, 290203, ...
  0, 1, 1, 47, 406, 5320, 65209, 876865, ...
  ...

Alois P. Heinz, Rows n = 0..23, flattened
Wikipedia, Integer intervals
Wikipedia, Iverson bracket
Wikipedia, Permanent (mathematics)
Wikipedia, Permutation
Wikipedia, Symmetric group

Columns k=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637.
Diagonals of the triangle (rows of the array) n=0-10 give: A002467 (for k>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630.
Row sums give A000142.
T(2n,n) or A(n,n) gives A324638.
Cf. A008305, A324564.

b:= proc(n, k) option remember; `if`(n=k, n!,
       LinearAlgebra[Permanent](Matrix(n, (i, j)->
      `if`(i<=j and j b(n, k)-`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);
# as array:
A:= (n, k)-> b(n+k, k)-`if`(k=0, 0, b(n+k, k-1)):
seq(seq(A(d-k, k), k=0..d), d=0..10);

b[n_, k_] := b[n, k] = If[n == k, n!, Permanent[Table[If[i <= j && j < k + i || n + j < k + i, 1, 0], {i, 1, n}, {j, 1, n}]]];
(* as triangle: *)
T[n_, k_] := b[n, k] - If[k == 0, 0, b[n, k - 1]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
(* as array: *)
A[n_, k_] := b[n + k, k] - If[k == 0, 0, b[n + k, k - 1]]; Table[A[d - k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* Jean-François Alcover, May 08 2019, after Alois P. Heinz *)

T(n,k) = |{ p in S_n : k = max_{i=1..n} (1+i-p(i)+n*[i0, T(0,0) = 1.

T(n,k) = A008305(n,k) - A008305(n,k-1) for k > 0, T(n,0) = A000007(n).

A170942 Take the permutations of lengths 1, 2, 3, ... arranged lexicographically, and replace each permutation with the number of its fixed points.

Original entry on oeis.org

1, 2, 0, 3, 1, 1, 0, 0, 1, 4, 2, 2, 1, 1, 2, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0, 5, 3, 3, 2, 2, 3, 3, 1, 2, 1, 1, 2, 2, 1, 3, 2, 1, 1, 1, 2, 2, 3, 1, 1, 3, 1, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 1, 0, 2, 1, 0, 0, 0, 1, 1, 2, 0, 0, 2, 0, 1, 0, 0, 1, 3, 1, 2, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Neven Juric (neven.juric(AT)apis-it.hr) and N. J. A. Sloane, Feb 23 2010

Length of n-th row = sum of n-th row = n!; number of zeros in n-th row = A000166(n); number of positive terms in n-th row = A002467(n). [Reinhard Zumkeller, Mar 29 2012]

			123,132,213,231,312,321 (corresponding to 3rd row of triangle A030298) have respectively 3,1,1,0,0,1 fixed points.

Reinhard Zumkeller, Rows n=1..7 of triangle, flattened
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation

Cf. A030298, A030299.
Cf. A008290, A000166, A000240, A000387, A000449, A000475, A129135, A129136, A129149, A129153, A129217, A129218, A129238, A129255.
Cf. A008291.

import Data.List (permutations, sort)
a170942 n k = a170942_tabf !! (n-1) (k-1)
a170942_row n = map fps $ sort $ permutations [1..n] where
   fps perm = sum $ map fromEnum $ zipWith (==) perm [1..n]
a170942_tabf = map a170942_row [1..]
-- Reinhard Zumkeller, Mar 29 2012

a(36)-a(105) from John W. Layman, Feb 23 2010

Keyword tabf added by Reinhard Zumkeller, Mar 29 2012

A068996 Decimal expansion of 1 - 1/e.

Original entry on oeis.org

6, 3, 2, 1, 2, 0, 5, 5, 8, 8, 2, 8, 5, 5, 7, 6, 7, 8, 4, 0, 4, 4, 7, 6, 2, 2, 9, 8, 3, 8, 5, 3, 9, 1, 3, 2, 5, 5, 4, 1, 8, 8, 8, 6, 8, 9, 6, 8, 2, 3, 2, 1, 6, 5, 4, 9, 2, 1, 6, 3, 1, 9, 8, 3, 0, 2, 5, 3, 8, 5, 0, 4, 2, 5, 5, 1, 0, 0, 1, 9, 6, 6, 4, 2, 8, 5, 2, 7, 2, 5, 6, 5, 4, 0, 8, 0, 3, 5, 6

Offset: 0

N. J. A. Sloane, Apr 08 2002

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, at least one person gets their own hat.
1-1/e is the limit to which (1 - !n/n!) {= 1 - A000166(n)/A000142(n) = A002467(n)/A000142(n)} converges as n tends to infinity. - Lekraj Beedassy, Apr 14 2005
Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is not connected. - Washington Bomfim, Nov 01 2010
Also equals the mode of a Gompertz distribution when the shape parameter is less than 1. - Jean-François Alcover, Apr 17 2013
The asymptotic density of numbers with an even number of trailing zeros in their factorial base representation (A232744). - Amiram Eldar, Feb 26 2021

			0.6321205588285576784044762...

Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, pp. 12-17.
Anders Hald, A History of Probability and Statistics and Their Applications before 1750, Wiley, NY, 1990 (Chapter 19).
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Brian Conrey and Tom Davis, Derangements.
Brady Haran and Tom Crawford, A Problem with Infinite Hotel Keys, YouTube Numberphile video, 2025.
MathOverflow, What is the effect of adding 1/2 to a continued fraction?.
Jonathan Sondow and Eric Weisstein, e, MathWorld.
Bala Subramanian, Why time constant is 63.2% not a 50 or 70%? (2018).
Index entries for transcendental numbers

Cf. A000166, A068985, A091131, A185393, A232744.

evalf[140](1-exp(-1));  # Alois P. Heinz, May 16 2025

RealDigits[1 - 1/E, 10, 100][[1]] (* Alonso del Arte, Jul 06 2012 *)

1 - exp(-1) \\ Michel Marcus, Mar 04 2016

Equals Integral_{x = 0 .. 1} exp(-x) dx. - Alonso del Arte, Jul 06 2012
Equals -Sum_{k>=1} (-1)^k/k!. - Bruno Berselli, May 13 2013
Equals Sum_{k>=0} 1/((2*k+2)*(2*k)!). - Fred Daniel Kline, Mar 03 2016
From Peter Bala, Nov 27 2019: (Start)
1 - 1/e =  Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n).
Continued fraction expansion: [0; 1, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...].
Related continued fraction expansions include
2*(1 - 1/e) = [1; 3, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, ..., 1, 3, 2*n + 1, 3, 1, 2*n + 1, ...];
(1/2)*(1 - 1/e) = [0; 3, 6, 10, 14, 18, ..., 4*n + 2, ...];
4*(1 - 1/e) = [2; 1, 1, 8, 3, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2, 2, 1, 1, 1, 3, ..., 7, 1, n, 2, 1, 1, 1, n+1, ...];
(1/4)*(1 - 1/e) = [0; 6, 3, 20, 7, 36, 11, 52, 15, ..., 16*n + 4, 4*n + 3, ...]. (End)
Equals Integral_{x=0..1} x * cosh(x) dx. - Amiram Eldar, Aug 14 2020
Equals A091131/e. - Hugo Pfoertner, Aug 20 2024

A306234 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n] divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 4, 3, 1, 1, 5, 13, 15, 13, 5, 1, 1, 7, 28, 67, 76, 67, 28, 7, 1, 1, 9, 49, 179, 411, 455, 411, 179, 49, 9, 1, 1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 23633, 10757, 3181, 679, 109, 13, 1

Offset: 1

Alois P. Heinz, Feb 17 2019

			Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    1,    1                        ;
  :                     1,    3,    4,    3,    1                  ;
  :               1,    5,   13,   15,   13,    5,   1             ;
  :          1,   7,   28,   67,   76,   67,   28,   7,  1         ;
  :      1,  9,  49,  179,  411,  455,  411,  179,  49,  9,  1     ;
  :  1, 11, 76, 375, 1306, 2921, 3186, 2921, 1306, 375, 76, 11, 1  ;

Alois P. Heinz, Rows n = 1..142, flattened
Wikipedia, Permutation

Columns k=0-10 give (offsets may differ): A002467, A180191, A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
Row sums give A306525.
T(n+1,n) gives A000012.
T(n+2,n) gives A005408.
T(n+2,n-1) gives A056107.
T(2n,n) gives A324361.
Cf. A000142, A306455, A306461, A324224, A324362.

b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
      add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, {})):
seq(T(n), n=1..8);
# second Maple program:
T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n)/abs(k)!:
seq(seq(T(n, k), k=1-n..n-1), n=1..9);

T[n_, k_] := (-1/Abs[k]!) Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
Table[T[n, k], {n, 1, 9}, {k, 1-n, n-1}] // Flatten (* Jean-François Alcover, Feb 15 2021 *)

T(n,k) = T(n,-k).
T(n,k) = -1/|k|! * Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.
T(n,k) = (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).
T(n+1,n) = 1.
T(n,k) = A306461(n,k) / |k|!.
Sum_{k=1-n..n-1} |k|! * T(n,k) = A306455(n).

A008305 Triangle read by rows: a(n,k) = number of permutations of [n] allowing i->i+j (mod n), j=0..k-1.

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 9, 24, 1, 2, 13, 44, 120, 1, 2, 20, 80, 265, 720, 1, 2, 31, 144, 579, 1854, 5040, 1, 2, 49, 264, 1265, 4738, 14833, 40320, 1, 2, 78, 484, 2783, 12072, 43387, 133496, 362880, 1, 2, 125, 888, 6208, 30818, 126565, 439792, 1334961, 3628800

Offset: 1

N. J. A. Sloane

The point is, we are counting permutations of [n] = {1,2,...,n} with the restriction that i cannot move by more than k places. Hence the phrase "permutations with restricted displacements". - N. J. A. Sloane, Mar 07 2014

The triangle could have been defined as an infinite square array by setting a(n,k) = n! for k >= n.

			a(4,3) = 9 because 9 permutations of {1,2,3,4} are allowed if each i can be placed on 3 positions i+0, i+1, i+2 (mod 4): 1234, 1423, 1432, 3124, 3214, 3412, 4123, 4132, 4213.
Triangle begins:
  1,
  1, 2,
  1, 2,   6,
  1, 2,   9,  24,
  1, 2,  13,  44,  120,
  1, 2,  20,  80,  265,   720,
  1, 2,  31, 144,  579,  1854,   5040,
  1, 2,  49, 264, 1265,  4738,  14833,  40320,
  1, 2,  78, 484, 2783, 12072,  43387, 133496,  362880,
  1, 2, 125, 888, 6208, 30818, 126565, 439792, 1334961, 3628800,
  ...

H. Minc, Permanents, Encyc. Math. #6, 1978, p. 48

Alois P. Heinz, Rows n = 1..23, flattened
Henry Beker and Chris Mitchell, Permutations with restricted displacement, SIAM J. Algebraic Discrete Methods 8 (1987), no. 3, 338--363. MR0897734 (89f:05009)
N. S. Mendelsohn, Permutations with confined displacement, Canad. Math. Bull., 4 (1961), 29-38.
N. Metropolis, M. L. Stein, P. R. Stein, Permanents of cyclic (0,1) matrices, J. Combin. Theory, 7 (1969), 291-321.
Wikipedia, Permanent (mathematics)

Diagonals (from the right): A000142, A000166, A000179, A000183, A004307, A189389, A184965.
Diagonals (from the left): A000211 or A048162, 4*A000382 or A004306 or A000803, A000804, A000805.
a(n,ceiling(n/2)) gives A306738.
Cf. A002467, A306714.

with(LinearAlgebra):
a:= (n, k)-> Permanent(Matrix(n,
             (i, j)-> `if`(0<=j-i and j-i

a[n_, k_] := Permanent[Table[If[0 <= j-i && j-i < k || j-i < k-n, 1, 0], {i, 1,n}, {j, 1, n}]]; Table[Table[a[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)

a(n,k) = per(sum(P^j, j=0..k-1)), where P is n X n, P[ i, i+1 (mod n) ]=1, 0's otherwise.

a(n,n) - a(n,n-1) = A002467(n). - Alois P. Heinz, Mar 06 2019

Comments and more terms from Len Smiley
More terms from Vladeta Jovovic, Oct 02 2003
Edited by Alois P. Heinz, Dec 18 2010

A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

Original entry on oeis.org

1, 0, 1, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 26, 32, 38, 46, 56, 66, 78, 92, 106, 123, 142, 162, 186, 214, 244, 280, 322, 368, 422, 484, 552, 630, 718, 815, 924, 1046, 1180, 1330, 1498, 1682, 1888, 2118, 2372, 2656, 2972, 3322, 3712, 4146, 4626

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(0) = 1 through a(12) = 12 partitions (A-C = 10..12; empty column indicated by dot; 0 is the empty partition):
   0  .  2  3    4    5    6    7    8     9     A      B      C
            21   31   41   51   43   53    54    64     65     75
                                61   71    63    73     74     84
                                     431   81    91     83     93
                                           432   532    A1     B1
                                           531   541    542    642
                                                 631    632    651
                                                 4321   641    732
                                                        731    741
                                                        5321   831
                                                               5421
                                                               6321

Alois P. Heinz, Table of n, a(n) for n = 0..5000

The version for permutations is A000166, complement A002467.
The reverse version is A025147, complement A238395, non-strict A238394.
The non-strict version is A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A238351, complement A352875.
The complement is A352829, non-strict A001522 (unproved, ranked by A352827 or A352874).
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352824, A352825, A352830, A352872.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]==0&]],{n,0,30}]

G.f.: Sum_{n>=0} q^(n*(3*n+1)/2)*Product_{k=1..n} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

A145877 Triangle read by rows: T(n,k) is the number of permutations of [n] for which the shortest cycle length is k (1<=k<=n).

Original entry on oeis.org

1, 1, 1, 4, 0, 2, 15, 3, 0, 6, 76, 20, 0, 0, 24, 455, 105, 40, 0, 0, 120, 3186, 714, 420, 0, 0, 0, 720, 25487, 5845, 2688, 1260, 0, 0, 0, 5040, 229384, 52632, 22400, 18144, 0, 0, 0, 0, 40320, 2293839, 525105, 223200, 151200, 72576, 0, 0, 0, 0, 362880, 25232230

Offset: 1

Emeric Deutsch, Oct 27 2008

Row sums are the factorials (A000142).
Sum(T(n,k), k=2..n) = A000166(n) (the derangement numbers).
T(n,1) = A002467(n).
T(n,n) = (n-1)! (A000142).
Sum(k*T(n,k),k=1..n) = A028417(n).
For the statistic "length of the longest cycle", see A126074.

			T(4,2)=3 because we have 3412=(13)(24), 2143=(12)(34) and 4321=(14)(23).
Triangle starts:
      1;
      1,    1;
      4,    0,    2;
     15,    3,    0,    6;
     76,   20,    0,    0, 24;
    455,  105,   40,    0,  0, 120;
   3186,  714,  420,    0,  0,   0, 720;
  25487, 5845, 2688, 1260,  0,   0,   0, 5040;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Cf. A000142, A000166, A002467, A028417, A126074.

T(2n,n) gives A110468(n-1) (for n>0). - Alois P. Heinz, Apr 21 2017

F:=proc(k) options operator, arrow: (1-exp(-x^k/k))*exp(-(sum(x^j/j, j = 1 .. k-1)))/(1-x) end proc: for k to 16 do g[k]:= series(F(k),x=0,16) end do: T:= proc(n,k) options operator, arrow: factorial(n)*coeff(g[k],x,n) end proc: for n to 11 do seq(T(n,k),k=1..n) end do; # yields sequence in triangular form

Rest[Transpose[Table[Range[0, 16]! CoefficientList[
      Series[(Exp[x^n/n] -1) (Exp[-Sum[x^k/k, {k, 1, n}]]/(1 - x)), {x, 0, 16}],x], {n, 1, 8}]]] // Grid (* Geoffrey Critzer, Mar 04 2011 *)

E.g.f. for column k is (1-exp(-x^k/k))*exp( -sum(j=1..k-1, x^j/j ) ) / (1-x). - Vladeta Jovovic

A324362 Total number of occurrences of k in the (signed) displacement sets of all permutations of [n+k] divided by k!; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 5, 13, 15, 0, 1, 7, 28, 67, 76, 0, 1, 9, 49, 179, 411, 455, 0, 1, 11, 76, 375, 1306, 2921, 3186, 0, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 0, 1, 15, 148, 1115, 6576, 29843, 98932, 214551, 229384, 0, 1, 17, 193, 1707, 12151, 69299, 307833, 1006007, 2160343, 2293839

Offset: 0

Alois P. Heinz, Feb 23 2019

			Square array A(n,k) begins:
    0,    0,     0,     0,     0,      0,      0, ...
    1,    1,     1,     1,     1,      1,      1, ...
    1,    3,     5,     7,     9,     11,     13, ...
    4,   13,    28,    49,    76,    109,    148, ...
   15,   67,   179,   375,   679,   1115,   1707, ...
   76,  411,  1306,  3181,  6576,  12151,  20686, ...
  455, 2921, 10757, 29843, 69299, 142205, 266321, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A002467, A180191(n+1), A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
Rows n=0-3 give: A000004, A000012, A005408, A056107(k+1).
Main diagonal gives A324361.
Cf. A306234.

A:= (n, k)-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!:
seq(seq(A(n, d-n), n=0..d), d=0..12);

m = 10;
col[k_] := col[k] = CoefficientList[(1-Exp[-x])/(1-x)^(k+1)+O[x]^(m+1), x]* Range[0, m]!;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 03 2021 *)

E.g.f. of column k: (1-exp(-x))/(1-x)^(k+1).
A(n,k) = -1/k! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+k-j)!.
A(n,k) = A306234(n+k,k).

A352829 Number of strict integer partitions y of n with a fixed point y(i) = i.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 26, 30, 36, 42, 50, 60, 70, 82, 96, 110, 126, 144, 163, 184, 208, 234, 264, 298, 336, 380, 430, 486, 550, 622, 702, 792, 892, 1002, 1125, 1260, 1408, 1572, 1752, 1950, 2168, 2408, 2672

Offset: 0

Gus Wiseman, May 15 2022

nonn

			The a(11) = 2 through a(17) = 12 partitions (A-F = 10..15):
  (92)   (A2)   (B2)    (C2)    (D2)     (E2)     (F2)
  (821)  (543)  (643)   (653)   (753)    (763)    (863)
         (921)  (A21)   (743)   (843)    (853)    (953)
                (5431)  (B21)   (C21)    (943)    (A43)
                        (5432)  (6432)   (D21)    (E21)
                        (6431)  (6531)   (6532)   (7532)
                                (7431)   (7432)   (7631)
                                (54321)  (7531)   (8432)
                                         (8431)   (8531)
                                         (64321)  (9431)
                                                  (65321)
                                                  (74321)

The non-strict version is A001522 (unproved, ranked by A352827 or A352874).
The version for permutations is A002467, complement A000166.
The reverse version is A096765 (or A025147 shifted right once).
The non-strict reverse version is A238395, ranked by A352872.
The complement is counted by A352828, non-strict A064428 (unproved, ranked by A352826 or A352873).
The version for compositions is A352875, complement A238351.
A000041 counts partitions, strict A000009.
A000700 counts self-conjugate partitions, ranked by A088902.
A008290 counts permutations by fixed points, unfixed A098825.
A115720 and A115994 count partitions by their Durfee square.
A238349 counts compositions by fixed points, complement A352523.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A352833 counts partitions by fixed points.
Cf. A008292, A064410, A111133, A114088, A118199, A188674, A257990, A352823, A352824, A352825, A352832.

pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&pq[#]>0&]],{n,0,30}]

G.f.: Sum_{n>=1} q^(n*(3*n-1)/2)*Product_{k=1..n-1} (1+q^k)/(1-q^k). - Jeremy Lovejoy, Sep 26 2022

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cp's OEIS Frontend

A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

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A324563 Number T(n,k) of permutations p of [n] such that k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+n*[i=0, 0<=k<=n, read by rows.

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A170942 Take the permutations of lengths 1, 2, 3, ... arranged lexicographically, and replace each permutation with the number of its fixed points.

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A068996 Decimal expansion of 1 - 1/e.

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A306234 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n] divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

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A008305 Triangle read by rows: a(n,k) = number of permutations of [n] allowing i->i+j (mod n), j=0..k-1.

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A352828 Number of strict integer partitions y of n with no fixed points y(i) = i.

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