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A105278 Triangle read by rows: T(n,k) = C(n,k)*(n-1)!/(k-1)!. 36
1, 2, 1, 6, 6, 1, 24, 36, 12, 1, 120, 240, 120, 20, 1, 720, 1800, 1200, 300, 30, 1, 5040, 15120, 12600, 4200, 630, 42, 1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1, 3628800, 16329600 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

T(n,k) is the number of partially ordered sets (posets) on n elements that consist entirely of k chains. For example, T(4, 3)=12 since there are exactly 12 posets on {a,b,c,d} that consist entirely of 3 chains. Letting ab denote a<=b and using a slash "/" to separate chains, the 12 posets can be given by a/b/cd, a/b/dc, a/c/bd, a/c/db, a/d/bc, a/d/cb, b/c/ad, b/c/da, b/d/ac, b/d/ca, c/d/ab and c/d/ba, where the listing of the chains is arbitrary (e.g., a/b/cd = a/cd/b =...cd/b/a). - Dennis P. Walsh, Feb 22 2007

Also the matrix product |S1|.S2 of Stirling numbers of both kinds.

This Lah triangle is a lower triangular matrix of the Jabotinsky type. See the column e.g.f. and the D. E. Knuth reference given in A008297. - Wolfdieter Lang, Jun 29 2007

The infinitesimal matrix generator of this matrix is given in A132710. See A111596 for an interpretation in terms of circular binary words and generalized factorials. - Tom Copeland, Nov 22 2007

Three combinatorial interpretations: T(n,k) is (1) the number of ways to split [n] = {1,..,n} into a collection of k nonempty lists ("partitions into sets of lists"), (2) the number of ways to split [n] into an ordered collection of n+1-k nonempty sets that are noncrossing ("partitions into lists of noncrossing sets"), (3) the number of Dyck n-paths with n+1-k peaks labeled 1,2,..n+1-k in some order. - David Callan, Jul 25 2008

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 = D where D(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 21 2008

An e.g.f. for the row polynomials of A(n,k) = T(n,k)*a(n-k) is exp[a(.)* D_x * x^2] exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,1)*a(.)*x], umbrally, where [(.)! Lag(.,x,1)]^n = n! Lag(n,x,1) is a normalized Laguerre polynomial of order 1. - Tom Copeland, Aug 29 2008

Triangle of coefficients from the Bell polynomial of the second kind for f=1/(1-x). B(n,k){x1,x2,x3,...}=B(n,k){1/(1-x)^2,...,(j-1)!/(1-x)^j,...}=T(n,k)/(1-x)^(n+k). - Vladimir Kruchinin, Mar 04 2011

The triangle, with the row and column offset taken as 0, is the generalized Riordan array (exp(x), x) with respect to the sequence n!*(n+1)! as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!^2 is A021009 unsigned). - Peter Bala, Aug 15 2013

For a relation to loop integrals in QCD, see p. 33 of Gopakumar and Gross and Blaizot and Nowak. - Tom Copeland, Jan 18 2016

Also the Bell transform of (n+1)!. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

LINKS

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened

J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.

Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5

J-P. Blaizot and M. Nowak, Large N_c confinement and turbulence, arXiv:0801.1859 [hep-th], 2008.

David Callan, Sets, Lists and Noncrossing Partitions , arXiv:0711.4841 [math.CO], 2007-2008.

P. Codara, O. M. D'Antona, P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.

T. Copeland, Mathemagical Forests, Addendum to Mathemagical Forests, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions, A Class of Differential Operators and the Stirling Numbers

S. Daboul, J. Mangaldan, M. Z. Spivey and P. Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., 86 (2013), 39-47.

G. H. E. Duchamp et al., Feynman graphs and related Hopf algebras, J. Phys. (Conf Ser) 30 (2006) 107-118.

R. Gopakumar and D. Gross, Mastering the master field, arXiv:hep-th/9411021, 1994.

G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4. - From Tom Copeland, Oct 01 2015

M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3

D. E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.

Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO]. - From N. J. A. Sloane, Aug 21 2012

MacTutor History of Mathematics archive: Biography of Ivo Lah.

Tilman Piesk, Illustration of the first four rows

W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.

Wikipedia, Lah number

FORMULA

T(n,k) = sum(|S1(n,m)|*S2(m,k),m=n..k), k>= n>=1, with Stirling triangles S2(n,m):=A048993 and S1(n,m):=A048994.

T(n,k) = C(n,k)*(n-1)!/(k-1)!.

Sum {k=1..n} T(n,k) = A000262(n).

n*Sum {k=1..n} T(n,k) = A103194(n) = Sum {k=1..n} T(n,k)*k^2 .

E.g.f. column k: (x^(k-1)/(1-x)^(k+1))/(k-1)!, k>=1.

Recurrence from Sheffer (here Jabotinsky) a-sequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n-1,m-1) + n*T(n-1,m). - Wolfdieter Lang, Jun 29 2007

The e.g.f. is, umbrally, exp[(.)!* L(.,-t,1)*x] = exp[t*x/(1-x)]/(1-x)^2 where L(n,t,1) = sum(k=0,...,n) T(n+1,k+1)*(-t)^k = sum(k=0,...,n) binomial(n+1,k+1)* (-t)^k / k! is the associated Laguerre polynomial of order 1. - Tom Copeland, Nov 17 2007

For this Lah triangle, the n-th row polynomial is given umbrally by

  n! C(B.(x)+1+n,n)= (-1)^n C(-B.(x)-2,n) , where C(x,n)=x!/(n!(x-n)!),

  the binomial coefficient, and B_n(x)= exp(-x)(xd/dx)^n exp(x), the n-th Bell / Touchard / exponential polynomial (cf. A008277). E.g.,

  2! C(-B.(-x)-2,2)=(-B.(x)-2)(-B.(x)-3)= B_2(x)+5*B_1(x)+6=6+6x+x^2.

  n! C(B.(x)+1+n,n)= n! e^(-x) Sum(j=0 to infin) C(j+1+n,n)x^j/j! is a corresponding Dobinski relation. See the Copeland link for the relation to inverse Mellin transform. - Tom Copeland, Nov 21 2011

The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A008277 (D = (1+x)*d/dx), A035342 (D = (1+x)^3*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). - Peter Bala, Nov 25 2011

T(n,k) = sum(A130534(n-1,i-1)*A008277(i,k), i=k..n). - Reinhard Zumkeller, Mar 18 2013

Let E(x) = sum {n >= 0} x^n/(n!*(n+1)!). Then a generating function is exp(t)*E(x*t) = 1 + (2 + x)*t + (6 + 6*x + x^2)*t^2/(2!*3!) + (24 + 36*x + 12*x^2 + x^3)*t^3/(3!*4!) + ... . - Peter Bala, Aug 15 2013

P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of A059110; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 23 2018

EXAMPLE

T(1,1) = C(1,1)*0!/0! = 1,

T(2,1) = C(2,1)*1!/0! = 2,

T(2,2) = C(2,2)*1!/1! = 1,

T(3,1) = C(3,1)*2!/0! = 6,

T(3,2) = C(3,2)*2!/1! = 6,

T(3,3) = C(3,3)*2!/2! = 1,

Sheffer a-sequence recurrence: T(6,2)= 1800 = (6/3)*120 +6*240.

B(n,k)=

1/(1-x)^2;

2/(1-x)^3, 1/(1-x)^4;

6/(1-x)^4,6/(1-x)^5,1/(1-x)^6;

24/(1-x)^5,36/(1-x)^6,12/(1-x)^7,1/(1-x)^8;

The triangle T(n,k) begins:

n\k      1       2       3      4      5     6    7  8  9 ...

1:       1

2:       2       1

3:       6       6       1

4:      24      36      12      1

5:     120     240     120     20      1

6:     720    1800    1200    300     30     1

7:    5040   15120   12600   4200    630    42    1

8:   40320  141120  141120  58800  11760  1176   56  1

9:  362880 1451520 1693440 846720 211680 28224 2016 72  1

...

Row n=10: [3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1]. - Wolfdieter Lang, Feb 01 2013

MAPLE

The triangle: for n from 1 to 13 do seq(binomial(n, k)*(n-1)!/(k-1)!, k=1..n) od;

the sequence: seq(seq(binomial(n, k)*(n-1)!/(k-1)!, k=1..n), n=1..13);

# The function BellMatrix is defined in A264428.

# Adds (1, 0, 0, 0, ..) as column 0.

BellMatrix(n -> (n+1)!, 9); # Peter Luschny, Jan 27 2016

MATHEMATICA

nn = 9; a = x/(1 - x); f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y a], {x, 0, nn}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 11 2011 *)

nn = 9; Flatten[Table[(j - k)! Binomial[j, k] Binomial[j - 1, k - 1], {j, nn}, {k, j}]] (* Jan Mangaldan, Mar 15 2013 *)

rows = 10;

t = Range[rows]!;

T[n_, k_] := BellY[n, k, t];

Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)

PROG

(Haskell)

a105278 n k = a105278_tabl !! (n-1) !! (k-1)

a105278_row n = a105278_tabl !! (n-1)

a105278_tabl = [1] : f [1] 2 where

   f xs i = ys : f ys (i + 1) where

     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))

-- Reinhard Zumkeller, Sep 30 2014, Mar 18 2013

(MAGMA) /* As triangle */ [[Binomial(n, k)*Factorial(n-1)/Factorial(k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 31 2014

(Perl) use ntheory ":all"; say join ", ", map { my $n=$_; map { stirling($n, $_, 3) } 1..$n; } 1..9; # Dana Jacobsen, Mar 16 2017

(GAP) Flat(List([1..10], n->List([1..n], k->Binomial(n, k)*Factorial(n-1)/Factorial(k-1)))); # Muniru A Asiru, Jul 25 2018

CROSSREFS

Cf. A000262, A103194, A105220.

Triangle of Lah numbers (A008297) unsigned.

Cf. |A111596(n, m)| (triangle with extra n=0 row and m=0 column).

Cf. A130561 (for a natural refinement).

Cf. A094638 (for differential operator representation).

Cf. A248045 (central terms), A002868 (row maxima).

Cf, A059110.

Sequence in context: A091599 A048999 A066667 * A008297 A090582 A079641

Adjacent sequences:  A105275 A105276 A105277 * A105279 A105280 A105281

KEYWORD

easy,nonn,tabl

AUTHOR

Miklos Kristof, Apr 25 2005

EXTENSIONS

Stirling comments and e.g.f.s from Wolfdieter Lang, Apr 11 2007

STATUS

approved

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Last modified September 24 20:09 EDT 2018. Contains 315356 sequences. (Running on oeis4.)