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A144696 Triangle of 2-Eulerian numbers. 8

%I

%S 1,1,2,1,7,4,1,18,33,8,1,41,171,131,16,1,88,718,1208,473,32,1,183,

%T 2682,8422,7197,1611,64,1,374,9327,49780,78095,38454,5281,128,1,757,

%U 30973,264409,689155,621199,190783,16867,256

%N Triangle of 2-Eulerian numbers.

%C Let [n] denote the ordered set {1,2,...,n}. The symmetric group S_n consists of the injective mappings p:[n] -> [n]. Such a permutation p has an excedance at position i, 1 <= i < n, if p(i) > i. One well-known interpretation of the Eulerian numbers A(n,k) is that they count the permutations in the symmetric group S_n with k excedances. The triangle of Eulerian numbers is A008292 (but with an offset of 1 in the column numbering). We generalize this definition to restricted permutations as follows.

%C Let r be a nonnegative integer and 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 p has an excedance at position i, 1 <= i <= n-r, if p(i) > i. Then the r-Eulerian number, denoted by A(r;n,k), is defined as the number of permutations in Permute(n,n-r) having k excedances. Thus the current array of 2-Eulerian numbers gives the number of permutations in Permute(n,n-2) with k excedances. See the example section below for some numerical examples.

%C Clearly A(0;n,k) = A(n,k). The case r = 1 also produces the ordinary Eulerian numbers A(n,k). There is an obvious bijection from Permute(n,n) to Permute(n,n-1) that preserves the number of excedances of a permutation. Consequently, the 1-Eulerian numbers are equal to the 0-Eulerian numbers: A(1;n,k) = A(0;n,k) = A(n,k).

%C For other cases of r-Eulerian numbers see A144697 (r = 3), A144698 (r = 4) and A144699 (r = 5). There is also a concept of r-Stirling numbers of the first and second kinds - see A143491 and A143494. If we multiply the entries of the current array by a factor of 2 and then reverse the rows we obtain A120434.

%C An alternative interpretation of the current array due to [Strosser] involves the 2-excedance statistic of a permutation (see also [Foata & Schutzenberger, Chapitre 4, Section 3]). We define a permutation p in Permute(n,n-2) to have a 2-excedance at position i (1 <=i <= n-2) if p(i) >= i + 2.

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

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

%D J. Riordan. - An introduction to combinatorial analysis. - New York, J. Wiley, 1958.

%D Carla D. Savage and Gopal Viswanathan, The 1/k-Eulerian Polynomials, Electr. J. Combinatorics, 19 (2012), #P9. - From _N. J. A. Sloane_, Feb 06 2013

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

%H J. F. Barbero G., J. Salas and E. J. S. Villase├▒or, <a href="http://arxiv.org/abs/1307.5624">Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications</a>, arXiv preprint arXiv:1307.5624, 2013

%H Mark Conger. <a href="http://arxiv.org/abs/math.CO/0508112">- A refinement of the Eulerian polynomials and the joint distribution of pi(1) and Des(pi) in S_n.</a>

%H D. Foata, M. Schutzenberger. <a href="http://www.arXiv.org/abs/math.CO/0508232">- Theorie Geometrique des Polynomes Euleriens</a>, Lecture Notes in Math., no.138, Springer Verlag 1970.

%H L. Liu, Y. Wang, <a href="http://www.arXiv.org/abs/math.CO/0509207">- A unified approach to polynomial sequences with only real zeros</a>

%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012

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

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

%F Recurrence relations:

%F 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 >= 2, T(2,k) = 0 for k >= 1. Special cases: T(n,n-2) = 2^(n-2); T(n,n-3) = A066810(n-1).

%F E.g.f. (with suitable offsets): 1/2*[(1 - x)/(1 - x*exp(t - t*x))]^2 = 1/2 + x*t + (x + 2*x^2)*t^2/2! + (x + 7*x^2 + 4*x^3)*t^3/3! + ... .

%F 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_2(x) = 1. It follows that the polynomials R_n(x) for n >= 3 have only real zeros (apply Corollary 1.2. of [Liu and Wang]).

%F The (n+1)-th row generating polynomial = 1/2!*Sum_{k = 1..n} (k+1)!*Stirling2(n,k)*x^(k-1)*(1-x)^(n-k).

%F For n >= 2,

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

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

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

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

%F Worpitzky-type identities:

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

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

%F Relation with Stirling numbers (Frobenius-type identities):

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

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

%F T(n+2,k) = 1/2! * Sum_{j = 0..n-k} (-1)^(n-k-j)*(j+2)!* binomial(n-j,k)*S(2;n+2,j+2) for n,k >= 0, where S(2;n,k) denotes the 2-Stirling numbers A143494(n,k).

%F The row polynomials of this array are related to the Eulerian polynomials. For example, 1/2*x*d/dx [x*(x + 4*x^2 + x^3)/(1-x)^4] = x^2*(1 + 7*x + 4*x^2)/(1-x)^5 and 1/2*x*d/dx [x*(x + 11*x^2 + 11*x^3 + x^4)/(1-x)^5] = x^2*(1 + 18*x + 33*x^2 + 8*x^3)/(1-x)^6.

%F Row sums A001710. Alternating row sums [1, -1, -2, 8, 16, -136, -272, 3968, 7936, ... ] are alternately (signed) tangent numbers and half tangent numbers - see A000182.

%F Sum_{k = 0..n-2} 2^k*T(n,k) = A069321(n-1). Sum_{k = 0..n-2} 2^(n-k)*T(n,k) = 4*A083410(n-1).

%F For n >=2, the shifted row polynomial t*R(n,t) = 1/2*D^(n-1)(f(x,t)) evaluated at x = 0, where D is the operator (1-t)*(1+x)*d/dx and f(x,t) = (1+x*t/(t-1))^(-2). - Peter Bala, Apr 22 2012

%e The triangle begins

%e ===========================================

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

%e ===========================================

%e 2..|..1

%e 3..|..1.....2

%e 4..|..1.....7.....4

%e 5..|..1....18....33.....8

%e 6..|..1....41...171...131....16

%e 7..|..1....88...718..1208...473....32

%e 8..|..1...183..2682..8422..7197..1611....64

%e ...

%e Row 4 = [1,7,4]: We represent a permutation p:[n-2] -> [n] in Permute(n,n-2) by its image vector (p(1),...,p(n-2)). Here n = 4. The permutation (1,2) has no excedances; 7 permutations have a single excedance, namely, (1,3), (1,4), (2,1), (3,1), (3,2), (4,1) and (4,2); the remaining 4 permutations, (2,3), (2,4), (3,4) and (4,3) each have two excedances.

%p with(combinat):

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

%p for n from 2 to 10 do

%p seq(T(n,k),k = 0..n-2)

%p end do;

%t T[n_, k_] := 1/2!*Sum[(-1)^(k - j)*Binomial[n + 1, k - j]*(j + 1)^(n - 1)* (j + 2), {j, 0, k}];

%t Table[T[n, k], {n, 2, 10}, {k, 0, n-2}] // Flatten (* _Jean-Fran├žois Alcover_, Oct 15 2019 *)

%Y Cf. A000182 (related to alt. row sums), A008292, A001710 (row sums), A120434, A143491, A143494, A143497, A144697, A144698, A144699.

%K easy,nonn,tabl

%O 2,3

%A _Peter Bala_, Sep 19 2008

%E Spelling/notation corrections by _Charles R Greathouse IV_, Mar 18 2010

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Last modified November 15 11:18 EST 2019. Contains 329144 sequences. (Running on oeis4.)