A002538 Second-order Eulerian numbers <>.

1, 8, 58, 444, 3708, 33984, 341136, 3733920, 44339040, 568356480, 7827719040, 115336085760, 1810992556800, 30196376985600, 532953524275200, 9927928075161600, 194677319705702400, 4008789120817152000, 86495828444928000000, 1951566265951948800000, 45958933902500720640000

Offset: 1

Author

N. J. A. Sloane, Simon Plouffe, Mira Bernstein, Robert G. Wilson v

Keywords

• nonn
• nice

Comments

Second-order Eulerian numbers <> count permutations of the multiset {1,1,2,2,...,n,n} with k ascents and the restriction that for any m <= n, all numbers between the two copies of m are less than m.

a(n) = number of edges in the Hasse diagram for the Bruhat order on permutations of [n+1]. - David Callan, Sep 03 2005

Proof. As explained on page 1 of the Stanley link, edges in the Hasse diagram of the (strong) Bruhat order on S_n are associated with pairs (pi,(i,j)) with pi in S_n and 1 <= i < j <= n, such that pi_i < pi_j and each entry of pi lying between pi_i and pi_j in POSITION does not lie between pi_i and pi_j in VALUE.

For example, pi = (3, 5, 1, 2, 4) gives edges for the (i,j) pairs (1,2), (1,5), (3,4), (4,5) but not, e.g., for (i,j) = (3,5) because 2 lies between pi_3=1 and pi_5=4 both in position and in value.

Let us count edges for a given pair (i,j). Consider the j-i+1 entries pi_i, pi_(i+1),...,pi_j. There are (j-i+1)! possible orderings for their values and (i,j) contributes an edge <=> the values of pi_i, pi_j are adjacent in this ordering with pi_i < pi_j.

There are (j-i)! such orderings (just coalesce the items pi_i, pi_j into a single item). The net result is that (i,j) contributes an edge 1/(j-i+1) of the time. So the total number of edges in the Hasse diagram is Sum_{1 <= i < j <= n} n!/(j-i+1) = (n+1)!(H_(n+1) - 2) + n! where H_n = 1 + 1/2 + 1/3 + ... + 1/n is the harmonic sum. QED - David Callan, Mar 07 2006

Number of reentrant corners along the lower contours of all deco polyominoes of height n+2. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. a(n) = Sum_{k>=1} k*A121579(n+2,k). - Emeric Deutsch, Aug 16 2006

a(n) is the total sum of the cycle maxima minus the cycle minima over all permutations of [n+1]. a(2) = 8 = 2+2+1+2+1+0: (123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3). - Alois P. Heinz, Dec 22 2023

a(n-1) is the number of parking functions of order n with exactly n-1 lucky cars, where a lucky car is a car which parks in the spot it prefers. - Kimberly P. Hadaway, Jun 20 2024

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd ed. Addison-Wesley, Reading, MA, 1994, p. 270.
• J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 83.
• 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

• Alois P. Heinz, Table of n, a(n) for n = 1..447 (first 100 terms from T. D. Noe)
• E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
• Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 4.
• L. Carlitz, Some numbers related to the Stirling numbers of the first and second kind, Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. Fiz., Numbers 544-576 (1976): 49-55. [Annotated scanned copy. The triangle is A008517.]
• Emma Colaric, Ryan DeMuse, Jeremy L. Martin, and Mei Yin, Interval parking functions, arXiv:2006.09321 [math.CO], 2020.
• I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33. (See Table 1.)
• O. J. Munch, Om potensproduktsummer [Norwegian, English summary], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. [Annotated scanned copy]
• O. J. Munch, Om potensproduktsummer [ Norwegian, English summary ], Nordisk Matematisk Tidskrift, 7 (1959), 5-19. There are errors in the last two rows of his table.
• Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.
• J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
• J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)
• R. P. Stanley, A survey of the Bruhat order of the symmetric group
• Albert Tarn, Approximations to certain square roots and the series of numbers connected therewith [Annotated scanned copy]

Crossrefs

Second diagonal of A008517 and second column of A112007.

Cf. A121579.

Programs

Magma

[n le 1 select n else (n+2)*Self(n-1) + n*Factorial(n): n in [1..30]]; // Vincenzo Librandi, Aug 11 2018

Maple

egf:= (x+2*log(1-x))/(x-1)^3:
a:= n-> n!*coeff(series(egf, x, n+1), x, n):
seq(a(n), n=1..21);  # Peter Luschny, Feb 12 2021
# Alternative:
a := n -> (n + 1)! * ((n + 2)*harmonic(n + 2) - 2*n - 3);
seq(a(n), n = 1..22);  # Peter Luschny, Apr 09 2024

Mathematica

Table[(-1)^(n + 1)* Sum[(-1)^(n - k) k (-1)^(n - k) StirlingS1[n + 3, k + 3], {k, 0, n}], {n, 1, 16}] (* Zerinvary Lajos, Jul 08 2009 *)
a[n_]:=(-1)*((2*n+3)*(n+1)!-Abs[StirlingS1[n+3,2]]);Flatten[Table[a[n],{n,1,21}]] (* Detlef Meya, Apr 09 2024 *)

PARI

N=66; x='x+O('x^66); Vec(serlaplace((x+2*log(1-x))/(x-1)^3)) \\ Joerg Arndt, Apr 09 2016

Formula

From Vladeta Jovovic, Sep 15 2003: (Start)

a(n) = Sum_{k=1..n} k * |Stirling1(n+2, k+2)|.

E.g.f.: (x+2*log(1-x))/(x-1)^3. (End)

With alternating signs: Ramanujan polynomials psi_2(n, x) evaluated at -1. - Ralf Stephan, Apr 16 2004

a(n) = (n+2)*a(n-1) + n*n!, n>=1, a(0):=0.

a(n) = (n+2)!*HarmonicSum(n+2) + (n+1)! - 2(n+2)! where HarmonicSum(n) = 1 + 1/2 + 1/3 + ... + 1/n. - David Callan, Mar 07 2006

a(n) = (n+1)!*((n+2)*h(n+2)-2*n-3) where h(n) = Sum_{k=1..n} 1/k. - Gary Detlefs, Mar 25 2011

Conjecture: a(n) + 2*(-n-2)*a(n-1) + (n^2+4*n+1)*a(n-2) - n*(n-1)*a(n-3) = 0. - R. J. Mathar, Oct 27 2014

a(n) = (-1)*((2*n + 3)*(n + 1)! - abs(Stirling1(n + 3, 2))). - Detlef Meya, Apr 09 2024

Extensions

More terms from Joerg Arndt, Apr 09 2016