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A105278
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Triangle read by rows: T(n,k) = C(n,k)*(n-1)!/(k-1)!.
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20
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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; internal format)
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OFFSET
| 1,2
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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 of 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 (dwalsh(AT)mtsu.edu), 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. - W. 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 (tcjpn(AT)msn.com), 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 (callan(AT)stat.wisc.edu), 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. [From Tom Copeland (tcjpn(AT)msn.com), 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. [From Tom Copeland (tcjpn(AT)msn.com), 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) [From Vladimir Kruchinin (kru(AT)ie.tusur.ru), Mar 04 2011]
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REFERENCES
| S. Daboul, J. Mangaldan, M. Z. Spivey and P. Taylor, The Lah Numbers and the nth Derivative of exp(1/x), http://math.pugetsound.edu/~mspivey/Exp.pdf.
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LINKS
| MacTutor History of Mathematics archive: Biography of Ivo Lah.
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, Arxiv preprint arXiv:1105.3043, 2011
David Callan, Sets, Lists and Noncrossing Partitions , arXiv:0711.4841
T. Copeland, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
G. H. E. Duchamp et al, Feynman graphs and related Hopf algebras, J. Phys. (Conf Ser) 30 (2006) 107-118
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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). W. 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 (tcjpn(AT)msn.com), Nov 17 2007
Contribution from Tom Copeland, Nov 21 2011: (Start)
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. (End)
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
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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;
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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);
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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 *)
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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).
Sequence in context: A091599 A048999 A066667 * A008297 A090582 A079641
Adjacent sequences: A105275 A105276 A105277 * A105279 A105280 A105281
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Miklos Kristof (kristmikl(AT)freemail.hu), Apr 25 2005
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EXTENSIONS
| Stirling comments and e.g.f.s from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Apr 11 2007.
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