

A105278


Triangle read by rows: T(n,k) = binomial(n,k)*(n1)!/(k1)!.


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
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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+1k nonempty sets that are noncrossing ("partitions into lists of noncrossing sets"), (3) the number of Dyck npaths with n+1k peaks labeled 1,2,...,n+1k in some order.  David Callan, Jul 25 2008
Given matrices A and B with A(n,k) = T(n,k)*a(nk) and B(n,k) = T(n,k)*b(nk), then A*B = D where D(n,k) = T(n,k)*[a(.)+b(.)]^(nk), umbrally.  Tom Copeland, Aug 21 2008
An e.g.f. for the row polynomials of A(n,k) = T(n,k)*a(nk) 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/(1x). B(n,k){x1,x2,x3,...} = B(n,k){1/(1x)^2,...,(j1)!/(1x)^j,...} = T(n,k)/(1x)^(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 kdimensional flats of the ndimensional 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+k1) in the basis of falling factorials (x)_k = x(x1)...(xk+1). Specifically, (x)^n = Sum_{k=1..n} T(n,k) (x)_k.  Jeremy L. Martin, Apr 21 2021


LINKS



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)*(n1)!/(k1)!.
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^(k1)/(1x)^(k+1))/(k1)!, k>=1.
Recurrence from Sheffer (here Jabotinsky) asequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n1,m1) + n*T(n1,m).  Wolfdieter Lang, Jun 29 2007
The e.g.f. is, umbrally, exp[(.)!* L(.,t,1)*x] = exp[t*x/(1x)]/(1x)^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 nth 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!(xn)!),
the binomial coefficient, and B_n(x)= exp(x)(xd/dx)^n exp(x), the nth 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
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 nth diagonal by n!, where the main diagonal is n=1, generates the Narayana matrix A001263.  Tom Copeland, Sep 23 2020


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 asequence recurrence: T(6,2)= 1800 = (6/3)*120 + 6*240.
B(n,k) =
1/(1x)^2;
2/(1x)^3, 1/(1x)^4;
6/(1x)^4, 6/(1x)^5, 1/(1x)^6;
24/(1x)^5, 36/(1x)^6, 12/(1x)^7, 1/(1x)^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)*(n1)!/(k1)!, k=1..n) od;
the sequence: seq(seq(binomial(n, k)*(n1)!/(k1)!, k=1..n), n=1..13);
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.


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];


PROG

(Haskell)
a105278 n k = a105278_tabl !! (n1) !! (k1)
a105278_row n = a105278_tabl !! (n1)
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]))
(Magma) /* As triangle */ [[Binomial(n, k)*Factorial(n1)/Factorial(k1): 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(n1)/Factorial(k1)))); # Muniru A Asiru, Jul 25 2018


CROSSREFS

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. A089231 (triangle with mirrored rows).
Cf. A271703 (triangle with extra n=0 row and m=0 column).


KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



