2,2

For a signed version of this triangle see A062137. The unsigned 2-restricted Lah number L(2;n,k) gives the number of partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1 and 2 must belong to different lists. More generally, the unsigned r-restricted Lah number L(r;n,k) gives the number of partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2, ..., r belong to different lists. If r = 1 there is no restriction and we obtain the unsigned Lah numbers A105278. For other cases see A143498 (r = 3) and A143499 (r = 4). We make some remarks on the general case.

The unsigned restricted Lah numbers occur as connection constants in the generalized Lah identity (x+2r-1)*(x+2r)*...*(x+2r+n-r-2) = sum {k = r..n} L(r;n,k)*(x-1)*(x-2)*...*(x-k+r) for n >=r and where any empty products are taken equal to 1 (for a bijective proof of the identity, follow the proof of [Petkovsek and Pisanski] but restrict r of the Argonauts to different paths).

The unsigned restricted Lah numbers satisfy the same recurrence as the unsigned Lah numbers, namely, L(r;n,k) = (n+k-1)*L(r;n-1,k) + L(r;n-1,k-1), but with the boundary conditions: L(r;n,k) = 0 if n < r or if k < r; L(r;r,r) = 1. The recurrence has the explicit solution L(r;n,k) = (n-r)!/(k-r)!*binomial(n+r-1,k+r-1) for n,k >= r. It follows that the unsigned r-restricted Lah numbers have 'vertical' generating functions for k >= r of the form sum {n >= k} L(r;n,k)*t^n/(n-r)! = 1/(k-r)!*t^k/(1-t)^(k+r). This yields the e.g.f. for the array of unsigned r-restricted Lah numbers in the form: sum {n,k >= r} L(r;n,k)*x^k*t^n/(n-r)! = (x*t)^r * 1/(1-t)^(2r) * exp(x*t/(1-t)) = (x*t)^r (1 + (2r+x)*t + (2r*(2r+1) + 2*(2r+1)*x + x^2)*t^2/2! + ... ). The array of unsigned r-restricted Lah numbers begins

1...................0..................0.............0...

2r..................1..................0.............0...

2r*(2r+1)...........2*(2r+1)...........1.............0...

2r*(2r+1)*(2r+2)....3*(2r+1)*(2r+2)....3*(2r+2)......1...

...

and equals exp(D(r)), where D(r) is the array with the sequence (2*r, 2*(2r+1), 3*(2r+2), 4*(2r+3), ... ) on the main subdiagonal and zeros everywhere else.

The unsigned r-restricted Lah numbers are related to the r-restricted Stirling numbers: the lower triangular array of unsigned r-restricted Lah numbers may be expressed as the matrix product St1(r) * St2(r), where St1(r) and St2(r) denote the arrays of r-restricted Stirling numbers of the first and second kind respectively. The theory of restricted Stirling numbers is developed in [Broder]. See A143491 - A143496 for tables of restricted Stirling numbers. An alternative factorization for the array is as St1 * P^(2r-2) * St2, where P denotes Pascal's triangle, A007318, St1 is the triangle of unsigned Stirling numbers of the first kind, abs(A008275) and St2 denotes the triangle of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).

The array of unsigned r-restricted Lah numbers is an example of the fundamental matrices sketched in A133314. So, redefining the offset as n=0, given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. An e.g.f. for the row polynomials of A is exp(x*t) exp{-x*t*[a*t/(a*t-1)]}/(1-a*t)^4 = exp(x*t) exp[(.)!*Laguerre(.,3,-x*t)*a(.)*t)], umbrally. [From Tom Copeland (tcjpn(AT)msn.com), Sep 19 2008]

Broder Andrei Z., The r-Stirling numbers, Discrete Math. 49, 241-259 (1984)

Neuwirth Erich, Recursively defined combinatorial functions: Extending Galton's board, Discrete Math. 239 No. 1-3, 33-51 (2001)

Marko Petkovsek, Tomaz Pisanski, Combinatorial interpretation of unsigned Stirling and Lah numbers

T(n,k) = (n-2)!/(k-2)!*C(n+1,k+1), n,k >= 2. Recurrence: T(n,k) = (n+k-1)*T(n-1,k) + T(n-1,k-1) for n,k >= 2, with the boundary conditions: T(n,k) = 0 if n < 2 or k < 2; T(2,2) = 1. E.g.f. for column k: sum {n >= k} T(n,k)*t^n/(n-2)! = 1/(k-2)!*t^k/(1-t)^(k+2) for k >= 2. E.g.f: sum {n = 2..inf} sum {k = 2..n} T(n,k)*x^k*t^n/(n-2)! = (x*t)^2/(1-t)^4*exp(x*t/(1-t)) = (x*t)^2*(1 + (4+x)t +(20+10x+x^2)t^2/2! + ... ). Generalized Lah identity: (x+3)*(x+4)*...*(x+n) = sum {k = 2..n} T(n,k)*(x-1)*(x-2)*...*(x-k+2). The polynomials 1/n!*sum {k = 2..n+2} T(n+2,k)*(-x)^(k-2) for n >= 0 are generalized Laguerre polynomials Laguerre(n,3,x). See A062137. Array = A143491 * A143494 = abs(A008275) * ( A007318 )^2 * A008277 (apply Theorem 10 of [Neuwirth]). Array equals exp(D), where D is the array with the quadratic sequence (4,10,18,28, ... ) on the main subdiagonal and zeros elsewhere.

After adding 1 to the head of the main diagonal and a zero to each of the subdiagonals, the n-th diagonal may be generated as coefficients of (1/n!) [D^(-1) tDt t^(-3)D t^3]^n exp(x t), where D is the derivative w.r.t. t and D^(-1) t^j/j! = t^(j+1)/(j+1)!. E.g., n=2 generates 20 x t^3/3! + 90 x^2 t^4/4! + 252 x^3 t^5/5! + ... . For the general unsigned r-restricted Lah number array, replace the threes by (2r-1) in the operator. The e.g.f. of the row polynomials is then exp[D^(-1) tDt t^(-(2r-1))D t^(2r-1)] exp(x t), with offset n=0. [From Tom Copeland (tcjpn(AT)msn.com), Sep 21 2008]

Triangle begins

n\k|.....2.....3.....4.....5.....6.....7

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

2..|.....1

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

4..|....20....10.....1

5..|...120....90....18.....1

6..|...840...840...252....28.....1

7..|..6720..8400..3360...560....40.....1

...

T(4,3) = 10. The partitions of {1,2,3,4} into 3 ordered lists such that the elements 1 and 2 lie in different lists are: {1}{2}{3,4} and {1}{2}{4,3}, {1}{3}{2,4} and {1}{3}{4,2}, {1}{4}{2,3} and {1}{4}{3,2}, {2}{3}{1,4} and {2}{3}{4,1}, {2}{4}{1,3} and {2}{4}{3,1}. The remaining two partitions {3}{4}{1,2} and {3}{4}{2,1} are not allowed because the elements 1 and 2 belong to the same block.

with combinat: T := (n, k) -> (n-2)!/(k-2)!*binomial(n+1, k+1): for n from 2 to 11 do seq(T(n, k), k = 2..n) end do;

Cf. A001715 (column 2), A007318, A008275, A008277, A061206 (column 3), A062137, A062141 - A062144 ( column 4 to column 7), A062146 (alt. row sums), A062147 (row sums), A105278 (unsigned Lah numbers), A143491, A143494, A143498, A143499.

Sequence in context: A049459 A143493 A062137 this_sequence A144354 A049352 A144484

Adjacent sequences: A143494 A143495 A143496 this_sequence A143498 A143499 A143500

easy,nonn,tabl

Peter Bala (pbala(AT)toucansurf.com), Aug 25 2008