A192721 The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1.

1, 3, 1, 19, 16, 1, 211, 299, 65, 1, 3651, 7346, 3156, 246, 1, 90921, 237517, 160322, 28722, 917, 1, 3081513, 9903776, 9302567, 2864912, 245407, 3424, 1, 136407699, 520507423, 632274183, 288196659, 46261609, 2041965, 12861, 1

Offset: 1

Author

Peter Bala, Jul 11 2011

Keywords

• nonn
• easy
• tabl

Comments

Let S_n denote the symmetric group on {1,2,...,n}. A permutation p_1p_2...p_n in S_n has a descent at position i (1 <= i <= n-1) if p_i > p_(i+1). The Eulerian numbers A008292 (with an offset of 0 in the column indexing) enumerate permutations by descents. We define a pair of permutations p_1p_2...p_n and q_1q_2...q_n to have a common descent at position i (1 <= i <= n-1) if both p_i > p_(i+1) and q_i > q_(i+1). For example, the permutations (3241) and (4231) in S_4 have common descents at positions i = 1 and i = 3. The table entry T(n,k) gives the number of pairs of permutations in the Cartesian product S_n x S_n with k common descents.

The generalized Stirling numbers associated with this triangle is A061691. See also A192722.

Examples

			The triangle begins
n/k|.....0.......1.......2......3....4.....5
============================================
..1|.....1
..2|.....3.......1
..3|....19......16.......1
..4|...211.....299......65......1
..5|..3651....7346....3156....246....1
..6|.90921..237517..160322..28722..917.....1
..
Row 3 entries T(3,0) = 19, T(3,1) = 16 and T(3,2) = 1 can be read from the following table:
============================================
Number of common descents in S_3 x S_3
============================================
.
...|.123...132...213...231...312...321
======================================
123|..0.....0.....0.....0.....0.....0
132|..0.....1.....0.....1.....0.....1
213|..0.....0.....1.....0.....1.....1
231|..0.....1.....0.....1.....0.....1
312|..0.....0.....1.....0.....1.....1
321|..0.....1.....1.....1.....1.....2
Matrix identity A192721 * A007318 = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....1.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|
		

Links

• Alois P. Heinz, Rows n = 1..45, flattened
• L. Carlitz, R. Scoville and T. Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc., 80 (1974), 881-884.
• L. Carlitz, R. Scoville, T. Vaughan, Enumeration of pairs of permutations, Discrete Math. 14, (1976) 215-239.
• J-Marc Fedou and D. Rawlings, More statistics on permutation pairs, The Electronic Journal of Combinatorics, 1 (1994) #R11.
• M. V. Koutras, Eulerian numbers associated with sequences of polynomials, Fibonacci Quart. 32 (1994) 44-57.
• T. Mendes, J. Remmel, A. Riehl, A Generalization of the Generating Functions for Descent Statistic.
• R. P. Stanley, Binomial posets, Möbius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.

Crossrefs

Cf. A000275 (first column), A001044 (row sums), A008292, A008459, A061691, A192722.

Programs

Maple

#A192721
#J = sum {n>=0} z^n/n!^2
J := unapply(BesselJ(0, 2*I*sqrt(z)),z):
G := (1-x)/(-x + J(z*(x-1))):
Gser := simplify(series(G, z = 0, 12)):
for n from 1 to 10 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 10 do seq(coeff(P[n],x,k), k = 0..n-1) od;
# gives sequence in triangular form

Mathematica

max = 9; j[z_] := BesselJ[0, 2 I*Sqrt[z]]; g = (1 - x)/(-x + j[z*(x - 1)]); gser = Series[g, {z, 0, max}]; p[n_] := n!^2 Coefficient[ gser, z, n]; a[n_, k_] := Coefficient[ p[n], x, k]; Flatten[ Table[ a[n, k], {n, 1, max-1}, {k, 0, n-1}]] (* Jean-François Alcover, Dec 13 2011, after Maple *)

Formula

Generating function (Carlitz et al. 1976): Let J(z) = sum {n>=0} z^n/n!^2. Then (1-x)/(J(z*(x-1))-x) = 1 + sum {n>=1} (sum {k = 0..n-1} T(n,k)*x^k)*z^n/n!^2 = 1 + z + (3+x)*z^2/2!^2 + (19+16*x+x^2)*z^3/3!^2 + .... Define a polynomial sequence {p(n,x) }n>=0 by means of the generating function J(z)^x = sum {n>=0} p(n,x)*z^n/n!^2. The generalized Eulerian polynomials associated with the sequence {p(n,x)} as defined by [Koutras, 1994] are the polynomials sum {k = 0..n-1} T(n,k)*x^(n-k).

Relations with other sequences: The first column of the array (x*I-A008459)^-1 (I the identity matrix) is a sequence of rational functions whose numerator polynomials are the row generating polynomials for the present triangle. The change of variable x -> (x+1)/x followed by z -> x*z transforms the above bivariate generating function (1-x)/(J(z*(x-1))-x) into 1/(1+x-x*J(z)), which is the generating function for A192722. Equivalently, if we postmultiply the present triangle by Pascal's triangle A007318 we obtain the row reversed form of A192722: A192721 * A007318 = row reverse of A192722.

Row n sum = n!^2 = A001044(n).

First column [1,3,19,211,3651,...] = A000275 (apart from initial term).