cp's OEIS Frontend

This is the case r = 3 of the r-Stirling numbers of the second kind. The 3-Stirling numbers of the second kind give the number of ways of partitioning the set {1,2,...,n} into k nonempty disjoint subsets with the restriction that the elements 1, 2 and 3 belong to distinct subsets. For remarks on the general case see A143494 (r = 2). The corresponding array of 3-Stirling numbers of the first kind is A143492. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For 3-Lah numbers refer to A143498.

From Peter Bala, Sep 19 2008: (Start)

Let D be the derivative operator d/dx and E the Euler operator x*d/dx. Then x^(-3)*E^n*x^3 = Sum_{k = 0..n} T(n+3,k+3)*x^k*D^k.

The row generating polynomials R_n(x) := Sum_{k= 3..n} T(n,k)*x^k satisfy the recurrence R_(n+1)(x) = x*R_n(x) + x*d/dx(R_n(x)) with R_3(x) = x^3. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).

Relation with the 3-Eulerian numbers E_3(n,j) := A144697(n,j): T(n,k) = (3!/k!)*Sum_{j = n-k..n-3} E_3(n,j)*binomial(j,n-k) for n >= k >= 3.

(End)

T(n,k) = S(n,k,3), n>=k>=3, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column k from (A20) with k->3, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(3*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(3*x)*((exp(x)-1)^m)/m!. See one of the formulas given below. For Sheffer matrices see the W. Lang link under A006232 with the S. Roman reference, also found in A132393. - Wolfdieter Lang, Sep 29 2011

The row polynomials s(n,x) := n!*L(n,5,x)= sum(a(n,m)*x^m,m=0..n) have e.g.f. exp(-z*x/(1-z))/(1-z)^6. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials sum(|A008297(n,m)|*(-x)^m, m=1..n), n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

These polynomials appear in the radial part of the l=2 (d-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.

For m=0..5 the (unsigned) column sequences (without leading zeros) are: A001725(n+5), A062148-A062152. Row sums (signed) give A062191; row sums (unsigned) give A062192.

The unsigned version of this triangle is the triangle of unsigned 3-Lah numbers A143498. - Peter Bala, Aug 25 2008

For a signed version of this triangle see A062137. The unsigned 2-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-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 r-Lah numbers occur as connection constants in the generalized Lah identity (x + 2*r - 1)*(x + 2*r)*...*(x + 2*r + 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 the first r of the Argonauts to different paths).

The unsigned r-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-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)^(2*r) * exp(x*t/(1 - t)) = (x*t)^r (1 + (2*r + x)*t + (2r*(2*r + 1) + 2*(2*r + 1)*x + x^2)*t^2/2! + ...).

The array of unsigned r-Lah numbers begins

1

2r 1

2r*(2r+1) 2*(2r+1) 1

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*(2*r + 1), 3*(2*r + 2), 4*(2*r + 3), ...) on the main subdiagonal and zeros everywhere else.

The unsigned r-Lah numbers are related to the r-Stirling numbers: the lower triangular array of unsigned r-Lah numbers may be expressed as the matrix product St1(r) * St2(r), where St1(r) and St2(r) denote the arrays of r-Stirling numbers of the first and second kind respectively. The theory of r-Stirling numbers is developed in [Broder]. See A143491 - A143496 for tables of r-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-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. - Tom Copeland, Sep 19 2008

See A049458 for a signed version of this array. The unsigned 3-Stirling numbers of the first kind count the permutations of the set {1,2,...,n} into k disjoint cycles, with the restriction that the elements 1, 2 and 3 belong to distinct cycles. This is the case r = 3 of the unsigned r-Stirling numbers of the first kind. For other cases see abs(A008275) (r = 1), A143491 (r = 2) and A143493 (r = 4). See A143495 for the corresponding 3-Stirling numbers of the second kind. The theory of r-Stirling numbers of both kinds is developed in [Broder]. For details of the related 3-Lah numbers see A143498.

With offset n=0 and k=0, this is the Sheffer triangle (1/(1-x)^3, -log(1-x)) (in the umbral notation of S. Roman's book this would be called Sheffer for (exp(-3*t), 1-exp(-t))). See the e.g.f given below. Compare also with the e.g.f. for the signed version A049458. - Wolfdieter Lang, Oct 10 2011

With offset n=0 and k=0 : triangle T(n,k), read by rows, given by (3,1,4,2,5,3,6,4,7,5,8,6,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 31 2011

This is the case r = 4 of the unsigned r-Lah numbers L(r;n,k). The unsigned 4-Lah numbers count the partitions of the set {1,2,...,n} into k ordered lists with the restriction that the elements 1, 2, 3 and 4 belong to different lists. For other cases see A105278 (r = 1), A143497 (r = 2 and comments on the general case) and A143498 (r = 3).

The unsigned 4-Lah numbers are related to the 4-Stirling numbers: the lower triangular array of unsigned 4-Lah numbers may be expressed as the matrix product St1(4) * St2(4), where St1(4) = A143493 and St2(4) = A143496 are the arrays of 4-restricted Stirling numbers of the first and second kind respectively. An alternative factorization for the array is as St1 * P^6 * 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.

The (2, 3)-Lah numbers T(n, k) count ordered 2-tuples (pi(1), pi(2)) of partitions of the set {1, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, 3 are in distinct lists, and bl(pi(1)) = bl(pi(2)) where for i = {1, 2} and pi(i) = b(1)^i, b(2)^i, ..., b(k)^i, where b(1)^i, b(2)^i, ..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min(b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).

The (2, 3)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=2 and r=3. More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} and pi(i) = {b(1)^i, b(2)^i, ..., b(k)^i}, where b(1)^i, b(2)^i,..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min (b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).

The (3, 3)-Lah numbers T(n, k) count ordered 3-tuples (pi(1), pi(2), pi(3)) of partitions of the set {1, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, 3 are in distinct lists, and bl(pi(1)) = bl(pi(2)) = bl(pi(3)) where for i = {1, 2, 3} and pi(i) = b(1)^i, b(2)^i, ..., b(k)^i, where b(1)^i, b(2)^i, ..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min(b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).

The (3, 3)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=3 and r=3. More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} and pi(i) = {b(1)^i, b(2)^i, ..., b(k)^i}, where b(1)^i, b(2)^i, ..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min (b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).