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A143494 Triangle read by rows: 2-Stirling numbers of the second kind. 26
1, 2, 1, 4, 5, 1, 8, 19, 9, 1, 16, 65, 55, 14, 1, 32, 211, 285, 125, 20, 1, 64, 665, 1351, 910, 245, 27, 1, 128, 2059, 6069, 5901, 2380, 434, 35, 1, 256, 6305, 26335, 35574, 20181, 5418, 714, 44, 1, 512, 19171, 111645, 204205, 156660, 58107, 11130, 1110, 54, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

This is the case r = 2 of the r-Stirling numbers of the second kind. The 2-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 and 2 belong to distinct subsets.

More generally, the r-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 numbers 1, 2, ..., r belong to distinct subsets. The case r = 1 gives the usual Stirling numbers of the second kind A008277; for other cases see A143495 (r = 3) and A143496 (r = 4).

The lower unitriangular array of r-Stirling numbers of the second kind equals the matrix product P^(r-1) * S (with suitable offsets in the row and column indexing), where P is Pascal's triangle, A007318 and S is the array of Stirling numbers of the second kind, A008277.

For the definition of and entries relating to the corresponding r-Stirling numbers of the first kind see A143491. For entries on r- Lah numbers refer to A143497. The theory of r-Stirling numbers of both kinds is developed in [Broder].

Contribution 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^(-2)*E^n*x^2 = sum {k = 0..n} T(n+2,k+2)*x^k*D^k.

The row generating polynomials R_n(x) := sum {k= 2..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_2(x) = x^2. It follows that the polynomials R_n(x) have only real zeros (apply Corollary 1.2. of [Liu and Wang]).

Relation with the 2-Eulerian numbers E_2(n,j) := A144696(n,j): T(n,k) = 2!/k!*sum {j = n-k..n-2} E_2(n,j)*binomial(j,n-k) for n >= k >= 2.

(End)

From Wolfdieter Lang, Sep 29 2011 (Start)

T(n,k)=S(n,k,2), n>=k>=2, in Mikhailov's first paper, eq.(28) or (A3). E.g.f. column no. k from (A20) with k->2, r->k. Therefore, with offset [0,0], this triangle is the Sheffer triangle (exp(2*x),exp(x)-1) with e.g.f. of column no. m>=0: exp(2*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.

(End)

REFERENCES

Michael J. Schlosser and Meesue Yoo, Elliptic Rook and File Numbers, Electronic Journal of Combinatorics, 24(1) (2017), #P1.31

LINKS

Table of n, a(n) for n=2..56.

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

C. B. Corcino, L. C. Hsu, E. L. Tan, Asymptotic approximations of r-Stirling numbers, Approximation Theory Appl. 15, No. 3 13-25 (1999)

A. Dzhumadildaev and D. Yeliussizov, Path decompositions of digraphs and their applications to Weyl algebra, arXiv preprint arXiv:1408.6764 [math.CO], 2014. [Version 1 contained many references to the OEIS, which were removed in Version 2. - N. J. A. Sloane, Mar 28 2015]

Askar Dzhumadil’daev and Damir Yeliussizov, Walks, partitions, and normal ordering, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006. [Peter Bala, Sep 19 2008]

V. V. Mikhailov, Ordering of some boson operator functions, J. Phys A: Math. Gen. 16 (1983) 3817-3827.

V. V. Mikhailov, Normal ordering and generalised Stirling numbers, J. Phys A: Math. Gen. 18 (1985) 231-235.

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

M. d’Ocagne, Sur une classe de nombres remarquables, Amer. J. Math., Vol. 9 (1887), 353-380.

M. Shattuck, Generalized r-Lah numbers, arXiv:1412.8721 [math.CO], 2014

FORMULA

T(n+2,k+2) = (1/k!)*sum {i = 0..k} (-1)^(k-i)*C(k,i)*(i+2)^n, n,k >= 0. T(n,k) = Stirling2(n,k) - Stirling2(n-1,k), n,k >= 2.

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

As a sum of monomial functions of degree m: T(n+m,n) = sum {2 <= i_1 <= ... <=i_m <=n} (i_1*i_2*...*i_m). For example, T(6,4) = sum {2<=i<=j<=4} (i*j) = 2*2 + 2*3 + 2*4 + 3*3 + 3*4 + 4*4 = 55.

E.g.f. column k+2 (with offset 2): 1/k!*exp(2x)*(exp(x)-1)^k.

O.g.f. k-th column: sum {n = k..inf} T(n,k)*x^n = x^k/((1-2*x)*(1-3*x)*...*(1-k*x)).

E.g.f.: exp(2*t + x*(exp(t)-1)) = sum {n = 0..inf} sum {k = 0..n} T(n+2,k+2) *x^k*t^n/n! = sum {n = 0..inf} B_n(2;x)*t^n/n! = 1 + (2 + x)*t/1! + (4 + 5*x + x^2)*t^2/2! + ..., where the row polynomial B_n(2;x) := sum {k = 0..n} T(n+2,k+2)*x^k denotes the 2-Bell polynomial.

Dobinski-type identities: Row polynomial B_n(2;x) = exp(-x)*sum {i = 0..inf} (i+2)^n*x^i/i!. Sum {k = 0..n} k!*T(n+2,k+2)*x^k = sum {i = 0..inf} (i+2)^n*x^i/(1+x)^(i+1).

The T(n,k) are the connection coefficients between falling factorials and the shifted monomials (x+2)^(n-2). For example, from row 4 we have 4 + 5*x + x*(x-1) = (x+2)^2, while from row 5 we have 8 + 19*x + 9*x*(x-1) + x*(x-1)*(x-2) = (x+2)^3.

The row sums of the array are the 2-Bell numbers, B_n(2;1), equal to A005493(n-2). The alternating row sums are the complementary 2-restricted Bell numbers, B_n(2;-1), equal to (-1)^n*A074051(n-2). This array is the matrix product P * S, where P denotes the Pascal triangle, A007318 and S denotes the lower triangular array of Stirling numbers of the second kind, A008277 (apply Theorem 10 of [Neuwirth]).

Also, this array equals the transpose of the upper triangular array A126351. The inverse array is A049444, the signed 2-Stirling numbers of the first kind. See A143491 for the unsigned version of the inverse.

Let f(x) = exp(exp(x)). Then for n >= 1, the row polynomials R(n,x) are given by R(n+2,exp(x)) = 1/f(x)*(d/dx)^n(exp(2*x)*f(x)). Similar formulas hold for A008277, A039755, A105794, A111577 and A154537. - Peter Bala, Mar 01 2012

EXAMPLE

Triangle begins

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

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

2..|...1

3..|...2....1

4..|...4....5....1

5..|...8...19....9....1

6..|..16...65...55...14....1

7..|..32..211..285..125...20....1

...

T(4,3) = 5. The set {1,2,3,4} can be partitioned into three subsets such that 1 and 2 belong to different subsets in 5 ways: {{1}{2}{3,4}}, {{1}{3}{2,4}}, {{1}{4}{2,3}}, {{2}{3}{1,4}} and {{2}{4}{1,3}}; the remaining possibility {{1,2}{3}{4}} is not allowed.

MAPLE

with combinat: T := (n, k) -> (1/(k-2)!)*add ((-1)^(k-i)*binomial(k-2, i)*(i+2)^(n-2), i = 0..k-2): for n from 2 to 11 do seq(T(n, k), k = 2..n) end do;

MATHEMATICA

t[n_, k_] := StirlingS2[n, k] - StirlingS2[n-1, k]; Flatten[ Table[ t[n, k], {n, 2, 11}, {k, 2, n}]] (* Jean-François Alcover, Dec 02 2011 *)

PROG

(Sage)

@CachedFunction

def stirling2r(n, k, r) :

    if n < r: return 0

    if n == r: return 1 if k == r else 0

    return stirling2r(n-1, k-1, r) + k*stirling2r(n-1, k, r)

A143494 = lambda n, k: stirling2r(n, k, 2)

for n in (2..6):

    [A143494(n, k) for k in (2..n)] # Peter Luschny, Nov 19 2012

CROSSREFS

Cf. A001047 (column 3), A005493 (row sums), A008277, A016269 (column 4), A025211 (column 5), A049444 (matrix inverse), A074051 (alt. row sums), A143491, A143495, A143496, A143497.

Sequence in context: A176667 A126182 A154342 * A124960 A137346 A264017

Adjacent sequences:  A143491 A143492 A143493 * A143495 A143496 A143497

KEYWORD

easy,nonn,tabl

AUTHOR

Peter Bala, Aug 20 2008

STATUS

approved

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Last modified November 17 00:02 EST 2018. Contains 317275 sequences. (Running on oeis4.)