A062119 -id:A062119 - cp's OEIS frontend

A001286 Lah numbers: a(n) = (n-1)*n!/2.

1, 6, 36, 240, 1800, 15120, 141120, 1451520, 16329600, 199584000, 2634508800, 37362124800, 566658892800, 9153720576000, 156920924160000, 2845499424768000, 54420176498688000, 1094805903679488000, 23112569077678080000, 510909421717094400000

Offset: 2

N. J. A. Sloane

Number of surjections from {1,...,n} to {1,...,n-1}. - Benoit Cloitre, Dec 05 2003
First Eulerian transform of 0,1,2,3,4,... . - Ross La Haye, Mar 05 2005
With offset 0 : determinant of the n X n matrix m(i,j)=(i+j+1)!/i!/j!. - Benoit Cloitre, Apr 11 2005
These numbers arise when expressing n(n+1)(n+2)...(n+k)[n+(n+1)+(n+2)+...+(n+k)] as sums of squares: n(n+1)[n+(n+1)] = 6(1+4+9+16+ ... + n^2), n(n+1)(n+2)(n+(n+1)+(n+2)) = 36(1+(1+4)+(1+4+9)+...+(1+4+9+16+ ... + n^2)), n(n+1)(n+2)(n+3)(n+(n+1)+(n+2)+(n+3)) = 240(...), ... . - Alexander R. Povolotsky, Oct 16 2006
a(n) is the number of edges in the Hasse diagram for the weak Bruhat order on the symmetric group S_n. For permutations p,q in S_n, q covers p in the weak Bruhat order if p,q differ by an adjacent transposition and q has one more inversion than p. Thus 23514 covers 23154 due to the transposition that interchanges the third and fourth entries. Cf. A002538 for the strong Bruhat order. - David Callan, Nov 29 2007
a(n) is also the number of excedances in all permutations of {1,2,...,n} (an excedance of a permutation p is a value j such p(j)>j). Proof: j is exceeded (n-1)! times by each of the numbers j+1, j+2, ..., n; now, Sum_{j=1..n} (n-j)(n-1)! = n!(n-1)/2. Example: a(3)=6 because the number of excedances of the permutations 123, 132, 312, 213, 231, 321 are 0, 1, 1, 1, 2, 1, respectively. - Emeric Deutsch, Dec 15 2008
(-1)^(n+1)*a(n) is the determinant of the n X n matrix whose (i,j)-th element is 0 for i = j, is j-1 for j>i, and j for j < i. - Michel Lagneau, May 04 2010
Row sums of the triangle in A030298. - Reinhard Zumkeller, Mar 29 2012
a(n) is the total number of ascents (descents) over all n-permutations. a(n) = Sum_{k=1..n} A008292(n,k)*k. - Geoffrey Critzer, Jan 06 2013
For m>=4, a(m-2) is the number of Hamiltonian cycles in a simple graph with m vertices which is complete, except for one edge. Proof: think of distinct round-table seatings of m persons such that persons "1" and "2" may not be neighbors; the count is (m-3)(m-2)!/2. See also A001710. - Stanislav Sykora, Jun 17 2014
Popularity of left (right) children in treeshelves. Treeshelves are ordered binary (0-1-2) increasing trees where every child is connected to its parent by a left or a right link. Popularity is the sum of a certain statistic (number of left children, in this case) over all objects of size n. See A278677, A278678 or A278679 for more definitions and examples. See A008292 for the distribution of the left (right) children in treeshelves. - Sergey Kirgizov, Dec 24 2016

			G.f. = x^2 + 6*x^3 + 36*x^4 + 240*x^5 + 1800*x^6 + 15120*x^7 + 141120*x^8 + ...
a(10) = (1+2+3+4+5+6+7+8+9)*(1*2*3*4*5*6*7*8*9) = 16329600. - _Reinhard Zumkeller_, May 15 2010

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 90, ex. 4.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 156.
A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 44.
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).

T. D. Noe, Table of n, a(n) for n = 2..100
Yasmin Aguillon et al., On Parking Functions and The Tower of Hanoi, arXiv:2206.00541 [math.CO], 2022.
Jean-Luc Baril, Sergey Kirgizov, and Vincent Vajnovszki, Patterns in treeshelves, arXiv:1611.07793 [cs.DM], 2016.
Kyle Celano, Jennifer Elder, Kimberly P. Hadaway, Pamela E. Harris, Jeremy L. Martin, Amanda Priestley, and Gabe Udell, Statistics on L-interval parking functions, arXiv:2507.07243 [math.CO], 2025. See p. 7.
Eldar Fischer, Johann A. Makowsky, and Vsevolod Rakita, MC-finiteness of restricted set partition functions, arXiv:2302.08265 [math.CO], 2023.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 399.
Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Martínez Mori, Boolean intervals in the weak order of S_n, arXiv:2306.14734 [math.CO], 2023.
Lucas Chaves Meyles, Pamela E. Harris, Richter Jordaan, Gordon Rojas Kirby, Sam Sehayek, and Ethan Spingarn, Unit-Interval Parking Functions and the Permutohedron, arXiv:2305.15554 [math.CO], 2023.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Sandi Klavžar, Uroš Milutinović and Ciril Petr, Hanoi graphs and some classical numbers, Expo. Math. 23 (2005), no. 4, 371-378.
Siegfried Lehr, Jeffrey Shallit, and John Tromp, On the vector space of the automatic reals, Theoret. Comput. Sci. 163 (1996), no. 1-2, 193-210.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257-260. [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Bruhat Graph.
Eric Weisstein's World of Mathematics, Edge Count.
Eric Weisstein's World of Mathematics, Permutation Ascent.

Cf. A001710, A052609, A062119, A075181, A060638, A060608, A060570, A060612, A135218, A019538, A053495, A051683, A213168, A278677, A278678, A278679, A008292.
A002868 is an essentially identical sequence.
Column 2 of |A008297|.
Third column (m=2) of triangle |A111596(n, m)|: matrix product of |S1|.S2 Stirling number matrices.
Cf. also A000110, A000111.
Cf. A001113, A001620, A091725, A099285.

a001286 n = sum[1..n-1] * product [1..n-1]
-- Reinhard Zumkeller, Aug 01 2011

[(n-1)*Factorial(n)/2: n in [2..25]]; // Vincenzo Librandi, Sep 09 2016

seq(sum(mul(j,j=3..n), k=2..n), n=2..21); # Zerinvary Lajos, Jun 01 2007

Table[Sum[n!, {i, 2, n}]/2, {n, 2, 20}] (* Zerinvary Lajos, Jul 12 2009 *)
nn=20;With[{a=Accumulate[Range[nn]],t=Range[nn]!},Times@@@Thread[{a,t}]] (* Harvey P. Dale, Jan 26 2013 *)
Table[(n - 1) n! / 2, {n, 2, 30}] (* Vincenzo Librandi, Sep 09 2016 *)

A001286(n):=(n-1)*n!/2$
makelist(A001286(n),n,1,30); /* Martin Ettl, Nov 03 2012 */

a(n)=(n-1)*n!/2 \\ Charles R Greathouse IV, Nov 20 2012

from _future_ import division
A001286_list = [1]
for n in range(2,100):
    A001286_list.append(A001286_list[-1]*n*(n+1)//(n-1)) # Chai Wah Wu, Apr 11 2018

[(n-1)*factorial(n)/2 for n in range(2, 21)] # Zerinvary Lajos, May 16 2009

a(n) = Sum_{i=0..n-1} (-1)^(n-i-1) * i^n * binomial(n-1,i). - Yong Kong (ykong(AT)curagen.com), Dec 26 2000 [corrected by Amiram Eldar, May 02 2022]
E.g.f.: x^2/[2(1-x)^2]. - Ralf Stephan, Apr 02 2004
a(n+1) = (-1)^(n+1)*det(M_n) where M_n is the n X n matrix M_(i,j)=max(i*(i+1)/2,j*(j+1)/2). - Benoit Cloitre, Apr 03 2004
Row sums of table A051683. - Alford Arnold, Sep 29 2006
5th binomial transform of A135218: (1, 1, 1, 25, 25, 745, 3145, ...). - Gary W. Adamson, Nov 23 2007
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j) then a(n)=(-1)^n*f(n,2,-2), (n>=2). - Milan Janjic, Mar 01 2009
a(n) = A000217(n-1)*A000142(n-1). - Reinhard Zumkeller, May 15 2010
a(n) = (n+1)!*Sum_{k=1..n-1} 1/(k^2+3*k+2). - Gary Detlefs, Sep 14 2011
Sum_{n>=2} 1/a(n) = 2*(2 - exp(1) - gamma + Ei(1)) = 1.19924064599..., where gamma = A001620 and Ei(1) = A091725. - Ilya Gutkovskiy, Nov 24 2016
a(n+1) = a(n)*n*(n+1)/(n-1). - Chai Wah Wu, Apr 11 2018
Sum_{n>=2} (-1)^n/a(n) = 2*(gamma - Ei(-1)) - 2/e, where e = A001113 and Ei(-1) = -A099285. - Amiram Eldar, May 02 2022

A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

Original entry on oeis.org

0, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
For n > 0, a(n) = n! - A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4) = 12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - Jon Perry, Jun 08 2003
With different offset, denominators of certain sums computed by Ramanujan.
From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n) = [2, 4, 12, 48, 240, ...] is A000629(n) = [2, 6, 26, 150, 1082, ...].
Stirling transform of a(n-1) = [1, 2, 4, 12, 48, ...] is A007047(n-1) = [1, 3, 11, 51, 299, ...].
Stirling transform of a(n) = [1, 4, 12, 48, 240, ...] is A002050(n) = [1, 5, 25, 149, 1081, ...].
Stirling transform of 2*A006252(n) = [2, 2, 4, 8, 28, ...] is a(n) = [2, 4, 12, 48, 240, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 2*A005649(n) = [4, 16, 88, 616, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 4*A083410(n) = [4, 16, 88, 616, ...]. (End)
Number of {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
Permanent of the (0, 1)-matrices with (i, j)-th entry equal to 0 if and only if it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix are the entries (i = 1, j = 1), (i = 1, j = n), (i = n, j = 1) and (i = n, j = n). - Simone Severini, Oct 17 2004

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 490.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 817.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 11.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for sequences related to factorial base representation.

Essentially the same sequence as A098558.
Cf. A062119, A080046, A245334, A000142.
Row 3 of A276955 (from term a(2)=4 onward).

a052849 n = if n == 0 then 0 else 2 * a000142 n
a052849_list = 0 : fs where fs = 2 : zipWith (*) [2..] fs
-- Reinhard Zumkeller, Aug 31 2014

[0] cat [2*Factorial(n-1): n in [2..25]]; // Vincenzo Librandi, Nov 03 2014

spec := [S,{B=Cycle(Z),C=Cycle(Z),S=Union(B,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Join[{0}, 2Range[20]!] (* Harvey P. Dale, Jul 13 2013 *)

PARI
```
a(n)=if(n<1,0,n!*2)
```

a(n) = T(n, 2) for n>1, where T is defined as in A080046.
D-finite with recurrence: {a(0) = 0, a(1) = 2, (-1 - n)*a(n+1) + a(n+2)=0}.
E.g.f.: 2*x/(1-x).
a(n) = A090802(n, n - 1) for n > 0. - Ross La Haye, Sep 26 2005
For n >= 1, a(n) = (n+3)!*Sum_{k=0..n+2} (-1)^k*binomial(2, k)/(n + 3 - k). - Milan Janjic, Dec 14 2008
G.f.: 2/Q(0) - 2, where Q(k) = 1 - x*(k + 1)/(1 - x*(k + 1)/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Apr 01 2013
G.f.: -2 + 2/Q(0), where Q(k) = 1 + k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: W(0) - 2 , where W(k) = 1 + 1/( 1 - x*(k+1)/( x*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n, n-1), n > 0. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=1} 1/a(n) = (e-1)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (e-1)/(2*e). (End)

More terms from Ross La Haye, Sep 26 2005

A257503 Square array A(row,col) read by antidiagonals: A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)); Dispersion of factorial base shift A255411 (array transposed).

Original entry on oeis.org

1, 2, 4, 3, 12, 18, 5, 16, 72, 96, 6, 22, 90, 480, 600, 7, 48, 114, 576, 3600, 4320, 8, 52, 360, 696, 4200, 30240, 35280, 9, 60, 378, 2880, 4920, 34560, 282240, 322560, 10, 64, 432, 2976, 25200, 39600, 317520, 2903040, 3265920, 11, 66, 450, 3360, 25800, 241920, 357840, 3225600, 32659200, 36288000, 13, 70, 456, 3456, 28800, 246240, 2540160, 3588480, 35925120, 399168000, 439084800

Offset: 1

Antti Karttunen, Apr 27 2015

The array is read by antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

The first row (A256450) contains all the numbers which have at least one 1-digit in their factorial base representation (see A007623), after which the successive rows are obtained from the terms on the row immediately above by shifting their factorial representation one left and then incrementing the nonzero digits in that representation with a factorial base shift-operation A255411.

			The top left corner of the array:
     1,     2,     3,     5,      6,      7,      8,      9,     10,     11,     13
     4,    12,    16,    22,     48,     52,     60,     64,     66,     70,     76
    18,    72,    90,   114,    360,    378,    432,    450,    456,    474,    498
    96,   480,   576,   696,   2880,   2976,   3360,   3456,   3480,   3576,   3696
   600,  3600,  4200,  4920,  25200,  25800,  28800,  29400,  29520,  30120,  30840
  4320, 30240, 34560, 39600, 241920, 246240, 272160, 276480, 277200, 281520, 286560
  ...

Transpose: A257505.
Inverse permutation: A257504.
Row index: A257679, Column index: A257681.
Row 1: A256450, Row 2: A257692, Row 3: A257693.
Columns 1-3: A001563, A062119, A130744 (without their initial zero-terms).
Column 4: A213167 (without the initial one).
Column 5: A052571 (without initial zeros).
Cf. A007623, A256450, A255411.
Cf. also permutations A255565 and A255566.
Thematically similar arrays: A083412, A135764, A246278.

(define (A257503 n) (A257503bi (A002260 n) (A004736 n)))
(define (A257503bi row col) (if (= 1 row) (A256450 (- col 1)) (A255411 (A257503bi (- row 1) col))))

A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A257505 Square array A(row,col): A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of factorial base shift A255411.

Original entry on oeis.org

1, 4, 2, 18, 12, 3, 96, 72, 16, 5, 600, 480, 90, 22, 6, 4320, 3600, 576, 114, 48, 7, 35280, 30240, 4200, 696, 360, 52, 8, 322560, 282240, 34560, 4920, 2880, 378, 60, 9, 3265920, 2903040, 317520, 39600, 25200, 2976, 432, 64, 10, 36288000, 32659200, 3225600, 357840, 241920, 25800, 3360, 450, 66, 11, 439084800, 399168000, 35925120, 3588480, 2540160, 246240, 28800, 3456, 456, 70, 13

Offset: 1

Antti Karttunen, Apr 27 2015

The array is read by downward antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

In Kimberling's terminology, this array is called the dispersion of sequence A255411 (when started from its first nonzero term, 4). The left column is the complement of that sequence, which is A256450.

			The top left corner of the array:
   1,   4,  18,   96,   600,   4320,   35280,   322560,   3265920
   2,  12,  72,  480,  3600,  30240,  282240,  2903040,  32659200
   3,  16,  90,  576,  4200,  34560,  317520,  3225600,  35925120
   5,  22, 114,  696,  4920,  39600,  357840,  3588480,  39553920
   6,  48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200
   7,  52, 378, 2976, 25800, 246240, 2575440, 29352960, 362517120
   8,  60, 432, 3360, 28800, 272160, 2822400, 31933440, 391910400
   9,  64, 450, 3456, 29400, 276480, 2857680, 32256000, 395176320
  10,  66, 456, 3480, 29520, 277200, 2862720, 32296320, 395539200
  11,  70, 474, 3576, 30120, 281520, 2898000, 32618880, 398805120
  13,  76, 498, 3696, 30840, 286560, 2938320, 32981760, 402433920
  14,  84, 552, 4080, 33840, 312480, 3185280, 35562240, 431827200
  15,  88, 570, 4176, 34440, 316800, 3220560, 35884800, 435093120
  17,  94, 594, 4296, 35160, 321840, 3260880, 36247680, 438721920
  19, 100, 618, 4416, 35880, 326880, 3301200, 36610560, 442350720
  20, 108, 672, 4800, 38880, 352800, 3548160, 39191040, 471744000
  21, 112, 690, 4896, 39480, 357120, 3583440, 39513600, 475009920
  23, 118, 714, 5016, 40200, 362160, 3623760, 39876480, 478638720
  ...

Antti Karttunen, Table of n, a(n) for n = 1..1275; the first 50 antidiagonals of array
Clark Kimberling, Interspersions and Dispersions.
Clark Kimberling, Interspersions and Dispersions, Proceedings of the American Mathematical Society, 117 (1993) 313-321.
Index entries for sequences related to factorial base representation
Index entries for sequences that are permutations of the natural numbers

Transpose: A257503.
Inverse permutation: A257506.
Row index: A257681, Column index: A257679.
Columns 1-3: A256450, A257692, A257693.
Rows 1-3: A001563, A062119, A130744 (without their initial zero-terms).
Row 4: A213167 (without the initial one).
Row 5: A052571 (without initial zeros).
Cf. A007623, A256450, A255411.
Cf. also permutations A255565, A255566.
Thematically similar arrays: A035513, A054582, A246279.

(define (A257505 n) (A257505bi (A002260 n) (A004736 n)))
(define (A257505bi row col) (if (= 1 col) (A256450 (- row 1)) (A255411 (A257505bi row (- col 1)))))

A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A067318 Sum of the reflection lengths of all permutations of n letters.

Original entry on oeis.org

0, 1, 7, 46, 326, 2556, 22212, 212976, 2239344, 25659360, 318540960, 4261576320, 61148511360, 937030429440, 15275952518400, 264030355814400, 4823280687052800, 92865738644582400, 1879691760950169600, 39905092126771200000, 886664974825728000000

Offset: 1

H. Nick Hann (nickhann(AT)aol.com), Jan 15 2002

The reflection length of a permutation is the minimum number of transpositions needed to express the permutation.
May also be called the "weight" of the symmetric group S_n.
a(n) is the number of n+1-permutations that have at least 3 cycles. a(n) = Sum_{k=3..n+1} A132393(n+1,k). Cf. A001563 (n-permutations with at least 2 cycles). - Geoffrey Critzer, Dec 01 2013
The number of covering relations in the middle order on S_n. - Bridget Tenner, Jul 11 2025

			a(3)=7 since the permutations are 1, (12), (13), (23), (12)(13) and (13)(12). The sum of reflection lengths of all elements in S_3 is 0+1+1+1+2+2=7.
The terms satisfy the series: x/(1-x) = x/((1+x)*(1+2*x)*(1+3*x)) + 7*x^2/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + 46*x^3/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) + 326*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)*(1+6*x)) + ... - _Paul D. Hanna_, Aug 28 2012

N. Hann, Average Weight of a Random Permutation, preprint, 2002. [Apparently unpublished]

G. C. Greubel, Table of n, a(n) for n = 1..445 (terms 1..100 from T. D. Noe).
Mathilde Bouvel, Luca Ferrari, and Bridget Eileen Tenner, Between weak and Bruhat: the middle order on permutations, Graphs and Combin., 41 (2025), #34.
Emma Colaric, Ryan DeMuse, Jeremy L. Martin, and Mei Yin, Interval parking functions, arXiv:2006.09321 [math.CO], 2020.
Walter Feit, Roger Lyndon, and Leonard L. Scott, A remark on permutations, Journal of Combinatorial Theory (A) 18 234-235 (1975).
Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 9.
Briana Foster-Greenwood and Cathy Kriloff, Spectra of Cayley Graphs of Complex Reflection Groups, arXiv preprint arXiv:1502.07392 [math.CO], 2015. (See remarks following Cor. 4.6.)
H. N. Hann, Symmetric Canonical Form [broken link]
Roberto Mantaci and Fanja Rakotondrajao, A permutation representation that knows what "Eulerian" means, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001).

Cf. A000254, A001563, A001705, A062119, A067369, A067370, A084938.

ZL :=[S, {S = Set(Cycle(Z),3 <= card)}, labelled]: seq(combstruct[count](ZL, size=n), n=2..22); # Zerinvary Lajos, Mar 25 2008

a[n_] := n!*(n - HarmonicNumber[n]); Table[a[n], {n, 1, 21}](* Jean-François Alcover, Feb 10 2012 *)
nn=22;Drop[Range[0,nn]!CoefficientList[Series[1/(1-x)-1-Log[1/(1-x)]-Log[1/(1-x)]^2/2!,{x,0,nn}],x],2] (* Geoffrey Critzer, Dec 01 2013 *)

A067318(n):=n*n! - abs(stirling1(n+1, 2))$
makelist(A067318(n),n,1,30); /* Martin Ettl, Nov 03 2012 */

{a(n)=if(n==0,0,if(n==1, 1, 1-polcoeff(sum(k=1, n-1, a(k)*x^k/prod(j=1, k+2, (1+j*x+x*O(x^n)) ) ), n)))} /* Paul D. Hanna, Aug 28 2012 */

a(n) = n!*(0/1+1/2+...+(n-1)/n) = n!*(n - H_n), where H_n = Sum_{k=1..n} 1/k; a(1) = 0, a(2) = 1, a(n) = n*a(n-1) + (n-1)*(n-1)!.
a(n) = n*n! - abs(stirling1(n+1, 2)) (cf. A000254). E.g.f.: (x+(1-x)*log(1-x))/(1-x)^2. - Vladeta Jovovic, Feb 01 2003
a(n) = T(n, n-1) for the triangle read by rows: [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 30 2003
G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n/Product_{k=1..n+2} (1+k*x). - Paul D. Hanna, Aug 28 2012
a(n) = A062119(n) - A001705(n-1). - Anton Zakharov, Sep 22 2016

Definition and example edited by Bridget Tenner, Jul 11 2025

A282466 a(n) = n*a(n-1) + n!, with n>0, a(0)=5.

Original entry on oeis.org

5, 6, 14, 48, 216, 1200, 7920, 60480, 524160, 5080320, 54432000, 638668800, 8143027200, 112086374400, 1656387532800, 26153487360000, 439378587648000, 7825123418112000, 147254595231744000, 2919482409811968000, 60822550204416000000, 1328364496464445440000

Offset: 0

Bruno Berselli, Feb 22 2017

C. Mariconda and A. Tonolo, Calcolo discreto, Apogeo (2012), page 240 (Example 9.57 gives the recurrence).

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A229039.

Cf. sequences with formula (n + k)*n!: A052521 (k=-5), A282822 (k=-4), A052520 (k=-3), A052571 (k=-2), A062119 (k=-1), A001563 (k=0), A000142 (k=1), A001048 (k=2), A052572 (k=3), A052644 (k=4), this sequence (k=5).

A282466:= func< n | (n+5)*Factorial(n) >; // G. C. Greubel, May 14 2025

RecurrenceTable[{a[0] == 5, a[n] == n a[n - 1] + n!}, a, {n, 0, 30}] (* or *)
Table[(n + 5) n!, {n, 0, 30}]

def A282466(n): return (n+5)*factorial(n) # G. C. Greubel, May 14 2025

E.g.f.: (5 - 4*x)/(1 - x)^2.
a(n) = (n + 5)*n!.
a(n) = 2*A229039(n) for n>0.
From Amiram Eldar, Nov 06 2020: (Start)
Sum_{n>=0} 1/a(n) = 9*e - 24.
Sum_{n>=0} (-1)^n/a(n) = 24 - 65/e. (End)

A066324 Number of endofunctions on n labeled points constructed from k rooted trees.

Original entry on oeis.org

1, 2, 2, 9, 12, 6, 64, 96, 72, 24, 625, 1000, 900, 480, 120, 7776, 12960, 12960, 8640, 3600, 720, 117649, 201684, 216090, 164640, 88200, 30240, 5040, 2097152, 3670016, 4128768, 3440640, 2150400, 967680, 282240, 40320, 43046721

Offset: 1

Christian G. Bower, Dec 14 2001

T(n,k) = number of endofunctions with k recurrent elements. - Mitch Harris, Jul 06 2006

The sum of row n is n^n, for any n. Basically the same sequence arises when studying random mappings (see A243203, A243202). - Stanislav Sykora, Jun 01 2014

			Triangle T(n,k) begins:
       1;
       2,      2;
       9,     12,      6;
      64,     96,     72,     24;
     625,   1000,    900,    480,   120;
    7776,  12960,  12960,   8640,  3600,   720;
  117649, 201684, 216090, 164640, 88200, 30240, 5040;
  ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 87, see (2.3.28).
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 3.3.32.

Alois P. Heinz, Rows n = 1..141, flattened

Column 1: A000169.
Main diagonal: A000142.
T(n, n-1): A062119.
Row sums give A000312.
Cf. A001864, A021010, A063169, A122525, A144084, A243203.

T:= (n, k)-> k*n^(n-k)*(n-1)!/(n-k)!:
seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 22 2012

f[list_] := Select[list, # > 0 &]; t = Sum[n^(n - 1) x^n/n!, {n, 1, 20}]; Flatten[Map[f, Drop[Range[0, 10]! CoefficientList[Series[1/(1 - y*t), {x, 0, 10}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 05 2011 *)

T(n, k)=k*n^(n-k)*(n-1)!/(n-k)! \\ Charles R Greathouse IV, Dec 05 2011

T(n,k) = k*n^(n-k)*(n-1)!/(n-k)!.
E.g.f. (relative to x): A(x, y)=1/(1-y*B(x)) - 1 = y*x +(2*y+2*y^2)*x^2/2! + (9*y+12*y^2+6*y^3)*x^3/3! + ..., where B(x) is e.g.f. A000169.
From Peter Bala, Sep 30 2011: (Start)
Let F(x,t) = x/(1+t*x)*exp(-x/(1+t*x)) = x*(1 - (1+t)*x + (1+4*t+2*t^2)*x^2/2! - ...). F is essentially the e.g.f. for A144084 (see also A021010). Then the e.g.f. for the present table is t*F(x,t)^(-1), where the compositional inverse is taken with respect to x.
Removing a factor of n from the n-th row entries results in A122525 in row reversed form.
(End)
Sum_{k=2..n} (k-1) * T(n,k) = A001864(n). - Geoffrey Critzer, Aug 19 2013
Sum_{k=1..n} k * T(n,k) = A063169(n). - Alois P. Heinz, Dec 15 2021

A156992 Triangle T(n,k) = n!*binomial(n-1, k-1) for 1 <= k <= n, read by rows.

Original entry on oeis.org

1, 2, 2, 6, 12, 6, 24, 72, 72, 24, 120, 480, 720, 480, 120, 720, 3600, 7200, 7200, 3600, 720, 5040, 30240, 75600, 100800, 75600, 30240, 5040, 40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320, 362880, 2903040, 10160640, 20321280, 25401600, 20321280, 10160640, 2903040, 362880

Offset: 1

Roger L. Bagula, Feb 20 2009

Partition {1,2,...,n} into m subsets, arrange (linearly order) the elements within each subset, then arrange the subsets. - Geoffrey Critzer, Mar 05 2010
From Dennis P. Walsh, Nov 26 2011: (Start)
Number of ways to arrange n different books in a k-shelf bookcase leaving no shelf empty.
There are n! ways to arrange the books in one long line. With ni denoting the number of books for shelf i, we have n = n1 + n2 + ... + nk. Since the number of compositions of n with k summands is binomial(n-1,k-1), we obtain T(n,k) = n!*binomial(n-1,k-1) for the number of ways to arrange the n books on the k shelves.
Equivalently, T(n,k) is the number of ways to stack n different alphabet blocks into k labeled stacks.
Also, T(n,k) is the number of injective functions f:[n]->[n+k] such that (i) the pre-image of (n+j) exists for j=1..k and (ii) f has no fixed points, that is, for all x, f(x) does not equal x.
T(n,k) is the number of labeled, rooted forests that have (i) exactly k roots, (ii) each root labeled larger than any nonroot, (iii) each root with exactly one child node, (iv) n non-root nodes, and (v) at most one child node for each node in the forest.
(End)
Essentially, the triangle given by (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) DELTA (2,1,3,2,4,3,5,4,6,5,7,6,8,7,9,8,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 29 2011
T(n,j+k) = Sum_{i=j..n-k} binomial(n,i)*T(i,j)*T(n-i,k). - Dennis P. Walsh, Nov 29 2011

			The triangle starts:
      1;
      2,      2;
      6,     12,      6;
     24,     72,     72,      24;
    120,    480,    720,     480,     120;
    720,   3600,   7200,    7200,    3600,    720;
   5040,  30240,  75600,  100800,   75600,  30240,   5040;
  40320, 282240, 846720, 1411200, 1411200, 846720, 282240, 40320;
From _Dennis P. Walsh_, Nov 26 2011: (Start)
T(3,2) = 12 since there are 12 ways to arrange books b1, b2, and b3 on shelves <shelf1><shelf2>:
   <b1><b2,b3>, <b1><b3,b2>, <b2><b1,b3>, <b2><b3,b1>,
   <b3><b1,b2>, <b3><b2,b1>, <b2,b3><b1>, <b3,b2><b1>,
   <b1,b3><b2>, <b3,b1><b2>, <b1,b2><b3>, <b2,b1><b3>.
(End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers

Cf. A002866 (row sums).
Column 1 = A000142. Column 2 = A001286 * 2! = A062119. Column 3 = A001754 * 3!. Column 4 = A001755 * 4!. Column 5 = A001777 * 5!. Column 6 = A001778 * 6!. Column 7 = A111597 * 7!. Column 8 = A111598 * 8!. Cf. A105278. - Geoffrey Critzer, Mar 05 2010
T(2n,n) gives A123072.

[Factorial(n)*Binomial(n-1,k-1): k in [1..n], n in [1..10]]; // G. C. Greubel, May 10 2021

seq(seq(n!*binomial(n-1,k-1),k=1..n),n=1..10); # Dennis P. Walsh, Nov 26 2011
with(PolynomialTools): p := (n,x) -> (n+1)!*hypergeom([-n],[],-x);
seq(CoefficientList(simplify(p(n,x)),x),n=0..5); # Peter Luschny, Apr 08 2015

Table[n!*Binomial[n-1, k-1], {n,10}, {k,n}]//Flatten

flatten([[factorial(n)*binomial(n-1,k-1) for k in (1..n)] for n in (1..10)]) # G. C. Greubel, May 10 2021

E.g.f. for column k is (x/(1-x))^k. - Geoffrey Critzer, Mar 05 2010
T(n,k) = A000142(n)*A007318(n-1,k-1). - Dennis P. Walsh, Nov 26 2011
Coefficient triangle of the polynomials p(n,x) = (n+1)!*hypergeom([-n],[],-x). - Peter Luschny, Apr 08 2015

A324225 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040

Offset: 1

Alois P. Heinz, Feb 18 2019

T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n].

			The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices
  [1    ]  [1    ]  [  1  ]  [  1  ]  [    1]  [    1]
  [  1  ]  [    1]  [1    ]  [    1]  [1    ]  [  1  ]
  [    1]  [  1  ]  [    1]  [1    ]  [  1  ]  [1    ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2].
Triangle T(n,k) begins:
  :                             1                           ;
  :                        1,   2,   1                      ;
  :                   2,   4,   6,   4,   2                 ;
  :              6,  12,  18,  24,  18,  12,   6            ;
  :        24,  48,  72,  96, 120,  96,  72,  48,  24       ;
  :  120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120  ;

Alois P. Heinz, Rows n = 1..100, flattened
Nadir Samos Sáenz de Buruaga, Rafał Bistroń, Marcin Rudziński, Rodrigo Miguel Chinita Pereira, Karol Życzkowski, and Pedro Ribeiro, Fidelity decay and error accumulation in quantum volume circuits, arXiv:2404.11444 [quant-ph], 2024. See p. 18.
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A062119, A052571, A052520, A282822, A052521.
Row sums give A001563.
T(n+1,n) gives A000142.
T(n+1,n-1) gives A052849.
T(n+1,n-2) gives A052560 for n>1.
Cf. A152883 (right half of this triangle without center column), A162608 (left half of this triangle), A306461, A324224.
Cf. A001710.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = |k|! * A324224(n,k).
E.g.f. of column k: x^t/t * hypergeom([2, t], [t+1], x) with t = |k|+1.
|T(n,k)-T(n,k-1)| = (n-1)! for k = 1-n..n.
Sum_{k=0..n-1} T(n,k) = A001710(n+1).

A233357 Triangle read by rows: T(n,k) = ((Stirling2)^2)(n,k) * k!

Original entry on oeis.org

1, 2, 2, 5, 12, 6, 15, 64, 72, 24, 52, 350, 660, 480, 120, 203, 2024, 5670, 6720, 3600, 720, 877, 12460, 48552, 83160, 71400, 30240, 5040, 4140, 81638, 424536, 983808, 1201200, 806400, 282240, 40320

Offset: 1

Tilman Piesk, Dec 07 2013

T(n,k) is the number of preferential arrangements with k levels of partitions of the set {1...n}.
2*T(n,k) is the number of formulas in first order logic that have an n-place predicate and k runs of A's and E's (universal and existential quantifiers, compare runs of 0's ans 1's counted by A005811), but don't include a negator.
4*T(n,k) is the number of such formulas that may include an negator.
T(n,k) is the number of partitions of an n-set into colored blocks, such that exactly k colors are used. T(3,2) = 12: 1a|23b, 1b|23a, 13a|2b, 13b|2a, 12a|3b, 12b|3a, 1a|2a|3b, 1b|2b|3a, 1a|2b|3a, 1b|2a|3b, 1a|2b|3b, 1b|2a|3a. - Alois P. Heinz, Sep 01 2019

			Let the colon ":" be a separator between two levels. E.g. in {1,2}:{3} the set {1,2} is on the first level, the set {3} is on the second level.
Compare descriptions of A083355 and A232598.
a(3,1)=5:
{1,2,3}
{1,2}{3}
{1,3}{2}
{2,3}{1}
{1}{2}{3}
a(3,2)=12:
{1,2}:{3}   {3}:{1,2}
{1,3}:{2}   {2}:{1,3}
{2,3}:{1}   {1}:{2,3}
{1}{2}:{3}   {3}:{1}{2}
{1}{3}:{2}   {2}:{1}{3}
{2}{3}:{1}   {1}:{2}{3}
a(3,3)=6:
{1}:{2}:{3}
{1}:{3}:{2}
{2}:{1}:{3}
{2}:{3}:{1}
{3}:{1}:{2}
{3}:{2}:{1}
Triangle begins:
       k = 1     2      3      4       5      6      7     8           sums
1          1                                                              1
2          2     2                                                        4
3          5    12      6                                                23
4         15    64     72     24                                        175
5         52   350    660    480     120                               1662
6        203  2024   5670   6720    3600    720                       18937
7        877 12460  48552  83160   71400  30240   5040               251729
8       4140 81638 424536 983808 1201200 806400 282240 40320        3824282

Tilman Piesk, First 100 rows, flattened
Tilman Piesk, Preferential arrangements of set partitions (Wikiversity)

A008277 (Stirling2), A039810 (square of Stirling2), A000110 (Bell), A000142 (factorials), A083355 (row sums: number of preferential arrangements), A232598 (number of preferential arrangements by number of blocks).

Cf. A130191.

S2 = A008277 (Stirling numbers of the second kind).
(S2)^2 = A039810 (matrix square of S2).
T(n,k) = ((S2)^2)(n,k) * k! = Sum(k<=i<=n) [ S2(n,i) * S2(i,k) ] * k!.
T(n,1) = Bell(n) = A000110(n).
T(n,2) = A052896(n).
T(n,n) = n! = A000142(n).
T(n,n-1) = n!*(n-1) = A062119(n).

cp's OEIS Frontend

A001286 Lah numbers: a(n) = (n-1)*n!/2.

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A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

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A257503 Square array A(row,col) read by antidiagonals: A(1,col) = A256450(col-1), and for row > 1, A(row,col) = A255411(A(row-1,col)); Dispersion of factorial base shift A255411 (array transposed).

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A257505 Square array A(row,col): A(row,1) = A256450(row-1), and for col > 1, A(row,col) = A255411(A(row,col-1)); Dispersion of factorial base shift A255411.

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A067318 Sum of the reflection lengths of all permutations of n letters.

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A282466 a(n) = n*a(n-1) + n!, with n>0, a(0)=5.

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