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 A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!. 44
 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 Also the number of k-dimensional flats of the n-dimensional Shi arrangement. - Shuhei Tsujie, Apr 26 2019 The numbers T(n,k) appear as coefficients when expanding the rising factorials (x)^k = x(x+1)...(x+k-1) in the basis of falling factorials (x)_k = x(x-1)...(x-k+1). Specifically, (x)^n = Sum_{k=1..n} T(n,k) (x)_k. - Jeremy L. Martin, Apr 21 2021 LINKS Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened J. Fernando Barbero G., Jesús Salas, and 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, and P. Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013. Tom 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 Milan 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. Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19. N. Nakashima and S. Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019. Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019. Tilman Piesk, Illustration of the first four rows Kornelia Ufniarz and Grzegorz Siudem, Combinatorial origins of the canonical ensemble, arXiv:2008.00244 [math-ph], 2020. W. Wang and T. Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500. Wikipedia, Lah number FORMULA T(n,k) = Sum_{m=n..k} |S1(n,m)|*S2(m,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} 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_{i=k..n} A130534(n-1,i-1)*A008277(i,k). - 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 Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates the Narayana matrix A001263. - Tom Copeland, Sep 23 2020 T(n,k) = A089231(n,n-k). - Ron L.J. van den Burg, Dec 12 2021 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 (signed 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. Cf. A001263, A008277. Cf. A089231 (triangle with mirrored rows). Cf. A271703 (triangle with extra n=0 row and m=0 column). 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 October 2 16:58 EDT 2023. Contains 365837 sequences. (Running on oeis4.)