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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

Examples

			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
		

References

• 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).

Links

• 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.

Crossrefs

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.

Programs

Haskell

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

Magma

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

Maple

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

Mathematica

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 *)

Maxima

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

PARI

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

Python

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

Sage

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

Formula

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