3,3

This is the case r = 3 of the r-restricted Eulerian numbers, denoted by A(r;n,k), defined as follows. Let [n] denote the ordered set {1,2,...,n} and let r be a nonnegative integer. Let Permute(n,n-r) denote the set of injective maps p:[n-r] -> [n], which we think of as permutations of n numbers taken n-r at a time. Clearly, |Permute(n,n-r)| = n!/r!. We say that the permutation p has an excedance at position i, 1 <= i <= n-r, if p(i) > i. Then the r-restricted Eulerian number A(r;n,k) is defined as the number of permutations in Permute(n,n-r) with k excedances. Thus the 3-restricted Eulerian numbers count the number of permutations in Permute(n,n-3) with k excedances (see the example section below for a numerical example).

For other cases see A008292 (r = 0 and r = 1), A144696 (r = 2), A144698 (r = 4) and A144699 (r = 5).

An alternative interpretation of the current array due to [Strosser] involves the 3-excedance statistic of a permutation (see also [Foata & Schutzenberger, Chapitre 4, Section 3]). We define a permutation p in Permute(n,n-3) to have a 3-excedance at position i (1 <=i <= n-3) if p(i) >= i + 3.

Given a permutation p in Permute(n,n-3), define ~p to be the permutation in Permute(n,n-3) that takes i to n+1 - p(n-i-2). The map ~ is a bijection of Permute(n,n-3) with the property that if p has (resp. does not have) an excedance in position i then ~p does not have (resp. has) a 3-excedance at position n-i-2. Hence ~ gives a bijection between the set of permutations with k excedances and the set of permutations with (n-k) 3-excedances. Thus reading the rows of this array in reverse order gives a triangle whose entries count the permutations in Permute(n,n-3) with k 3-excedances.

Example: Represent a permutation p:[n-3] -> [n] in Permute(n,n-3) by its image vector (p(1),...,p(n-3)). In Permute(10,7) the permutation p = (1,2,4,10,3,6,5) does not have an excedance in the first two positions (i = 1 and 2) or in the final three positions (i = 5, 6 and 7). The permutation ~p = (6,5,8,1,7,9,10) has 3-excedances only in the first three positions and the final two positions.

R. Strosser. - Seminaire de theorie combinatoire, I.R.M.A., Universite de Strasbourg, 1969-1970.

Mark Conger. - A refinement of the Eulerian polynomials and the joint distribution of pi(1) and Des(pi) in S_n.

D. Foata, M. Schutzenberger. - Theorie Geometrique des Polynomes Euleriens, Lecture Notes in Math., no.138, Springer Verlag 1970.

L. Liu, Y. Wang, - A unified approach to polynomial sequences with only real zeros

T(n,k) = 1/3!*sum {j = 0..k} (-1)^(k-j)*binomial(n+1,k-j)*(j+1)^(n-2)*(j+2)*(j+3);

T(n,n-k) = 1/3!*sum {j = 3..k} (-1)^(k-j)*binomial(n+1,k-j)*j^(n-2)*(j-1)*(j-2).

Recurrence relation:

T(n,k) = (k+1)*T(n-1,k) + (n-k)*T(n-1,k-1) with boundary conditions T(n,0) = 1 for n >= 3, T(3,k) = 0 for k >= 1. Special cases: T(n,n-3) = 3^(n-3); T(n,n-4) = A086443 (n-2).

E.g.f. (with suitable offsets): 1/3*[(1 - x)/(1 - x*exp(t - t*x))]^3 = 1/3 + x*t + (x + 3*x^2)*t^2/2! + (x + 10*x^2 + 9*x^3)*t^3/3! + ... . The row generating polynomials R_n(x) satisfy the recurrence R_(n+1)(x) = (n*x+1)*R_n(x) + x*(1-x)*d/dx(R_n(x)) with R_3(x) = 1. It follows that the polynomials R_n(x) for n >= 4 have only real zeros (apply Corollary 1.2. of [Liu and Wang]). The (n+2)-th row generating polynomial = 1/3!*sum {k = 1..n} (k+2)!*Stirling2(n,k) *x^(k-1)*(1-x)^(n-k).

For n >= 3,

1/3*(x*d/dx)^(n-2) (1/(1-x)^3) = x/(1-x)^(n+1) * sum {k = 0..n-3} T(n,k)*x^k,

1/3*(x*d/dx)^(n-2) (x^3/(1-x)^3) = 1/(1-x)^(n+1) * sum {k = 3..n} T(n,n-k)*x^k,

1/(1-x)^(n+1) * sum {k = 0..n-3} T(n,k)*x^k = 1/3! * sum {m = 0..inf} (m+1)^(n-2)*(m+2)*(m+3)*x^m,

1/(1-x)^(n+1) * sum {k = 3..n} T(n,n-k)*x^k = 1/3! * sum {m = 3..inf} m^(n-2)*(m-1)*(m-2)*x^m.

Worpitzky-type identities:

sum {k = 0..n-3} T(n,k)* binomial(x+k,n) = 1/3!*x^(n-2)*(x-1) *(x-2);

sum {k = 3..n} T(n,n-k)* binomial(x+k,n) = 1/3!*(x+1)^(n-2)*(x+2)*(x+3).

Relation with Stirling numbers (Frobenius-type identities):

T(n+2,k-1) = 1/3! * sum {j = 0..k} (-1)^(k-j)* (j+2)!*binomial(n-j,k-j)*Stirling2(n,j) for n,k >= 1;

T(n+2,k-1) = 1/3! * sum {j = 0..n-k} (-1)^(n-k-j)* (j+2)!*binomial(n-j,k)*S(3;n+3,j+3) for n,k >= 1 and

T(n+3,k) = 1/3! * sum {j = 0..n-k} (-1)^(n-k-j)*(j+3)!*binomial(n-j,k)*S(3;n+3,j+3) for n,k >= 0, where S(3;n,k) denotes the 3-restricted Stirling numbers A143495(n,k).

The row polynomials of this array are related to the 2-restricted Eulerian polynomials (see A144696). For example, 1/3*x*d/dx [x^3*(1 + 7*x + 4*x^2)/(1-x)^5] = x^3*(1 + 10*x + 9*x^2)/(1-x)^6 and 1/3*x*d/dx [x^3*(1 + 18*x + 33*x^2 + 8*x^3)/(1-x)^6] = x^3*(1 + 25*x + 67*x^2 + 27*x^3)/(1-x)^7.

Triangle begins

=================================================

n\k|..0......1......2......3......4......5......6

=================================================

3..|..1

4..|..1......3

5..|..1.....10......9

6..|..1.....25.....67.....27

7..|..1.....56....326....376.....81

8..|..1....119...1314...3134...1909....243

9..|..1....246...4775..20420..25215...9094....729

...

T(5,1) = 10: We represent a permutation p:[n-3] -> [n] in Permute(n,n-3) by its image vector (p(1),...,p(n-3)). The 10 permutations in Permute(5,2) having 1 excedance are (1,3), (1,4), (1,5), (3,2), (4,2), (5,2), (2,1), (3,1), (4,1) and (5,1).

with(combinat):

T:= (n, k) -> 1/3!*add((-1)^(k-j)*binomial(n+1, k-j)*(j+1)^(n-2)*(j+2)*(j+3), j = 0..k):

for n from 3 to 11 do

seq(T(n, k), k = 0..n-3)

end do;

A001715 (row sums), A008292, A060056, A143492, A143495, A144696, A144698, A144699.

Sequence in context: A124574 A052964 A084178 this_sequence A146154 A068438 A064060

Adjacent sequences: A144694 A144695 A144696 this_sequence A144698 A144699 A144700

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Sep 19 2008