The OEIS is supported by the many generous donors to the OEIS Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A008279 Triangle T(n,k) = n!/(n-k)! (0 <= k <= n) read by rows, giving number of permutations of n things k at a time. 114
 1, 1, 1, 1, 2, 2, 1, 3, 6, 6, 1, 4, 12, 24, 24, 1, 5, 20, 60, 120, 120, 1, 6, 30, 120, 360, 720, 720, 1, 7, 42, 210, 840, 2520, 5040, 5040, 1, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 1, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Also called permutation coefficients. Also falling factorials triangle A068424 with column a(n,0)=1 and row a(0,1)=1 otherwise a(0,k)=0, added. - Wolfdieter Lang, Nov 07 2003 The higher-order exponential integrals E(x,m,n) are defined in A163931; for information about the asymptotic expansion of E(x,m=1,n) see A130534. The asymptotic expansions for n = 1, 2, 3, 4, ..., lead to the right hand columns of the triangle given above. - Johannes W. Meijer, Oct 16 2009 The number of injective functions from a set of size k to a set of size n. - Dennis P. Walsh, Feb 10 2011 The number of functions f from {1,2,...,k} to {1,2,...,n} that satisfy f(x) >= x for all x in {1,2,...,k}. - Dennis P. Walsh, Apr 20 2011 T(n,k) = A181511(n,k) for k=1..n-1. - Reinhard Zumkeller, Nov 18 2012 The e.g.f.s enumerating the faces of the permutohedra / permutahedra, Perm(s,t;x) = [e^(sx)-1]/[s-t(e^(sx)-1)], (cf. A090582 and A019538) and the stellahedra / stellohedra, St(s,t;x) = [s e^((s+t)x)]/[s-t(e^(sx)-1)], (cf. A248727) given in Toric Topology satisfy exp[u*d/dt] St(s,t;x) = St(s,u+t;x) = [e^(ux)/(1-u*Perm(s,t;x))]*St(s,t;x), where e^(ux)/(1-uy) is a bivariate e.g.f. for the row polynomials of this entry and A094587. Equivalently, d/dt St = (x+Perm)*St and d/dt Perm = Perm^2, or d/dt log(St) = x + Perm and d/dt log(Perm) = Perm. - Tom Copeland, Nov 14 2016 T(n, k)/n! are the coefficients of the n-th exponential Taylor polynomial, or truncated exponentials, which was proved to be irreducible by Schur. See Coleman link. - Michel Marcus, Feb 24 2020 REFERENCES CRC Standard Mathematical Tables and Formulae, 30th ed., 1996, p. 176; 31st ed., p. 215, Section 3.3.11.1. Maple V Reference Manual, p. 490, numbperm(n,k). LINKS T. D. Noe, Rows n = 0..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], 2013-2014. V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014. R. F. Coleman, On the Galois groups of the exponential Taylor polynomials, L’Enseignement Mathématique 33, 183-189, (1987). J. Goldman, J. Haglund, Generalized rook polynomials, J. Combin. Theory A91 (2000), 509-530, 2-rook coefficients for k rooks on the 1xn board, all heights 1. R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994. Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963) 31-41. Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4. T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy] OEIS Wiki, Sorting numbers Yuriy Shablya, Dmitry Kruchinin, Euler-Catalan's Number Triangle and its Application, Proceedings Book of Micropam (The Mediterranean International Conference of Pure & Applied Mathematics and Related Areas 2018), 212-215. Dennis Walsh, Notes on no-less functions Eric Weisstein's World of Mathematics, Falling Factorial FORMULA E.g.f.: Sum T(n,k) x^n/n! y^k = exp(x)/(1-x*y). - Vladeta Jovovic, Aug 19 2002 Equals A007318 * A136572. - Gary W. Adamson, Jan 07 2008 T(n, k) = n*T(n-1, k-1) = k*T(n-1, k-1)+T(n-1, k) = n*T(n-1, k)/(n-k) = (n-k+1)*T(n, k-1). - Henry Bottomley, Mar 29 2001 T(n, k) = n!/(n-k)! if n >= k >= 0, otherwise 0. G.f. for k-th column k!*x^k/(1-x)^(k+1), k >= 0. E.g.f. for n-th row (1+x)^n, n >= 0. Sum T(n, k)x^k = permanent of n X n matrix a_ij = (x+1 if i=j, x otherwise). - Michael Somos, Mar 05 2004 Ramanujan psi_1(k, x) polynomials evaluated at n+1. - Ralf Stephan, Apr 16 2004 E.g.f.: Sum T(n,k) x^n/n! y^k/k! = e^{x+xy}. - Franklin T. Adams-Watters, Feb 07 2006 The triangle is the binomial transform of an infinite matrix with (1, 1, 2, 6, 24, ...) in the main diagonal and the rest zeros. - Gary W. Adamson, Nov 20 2006 G.f.: 1/(1-x-xy/(1-xy/(1-x-2xy/(1-2xy/(1-x-3xy/(1-3xy/(1-x-4xy/(1-4xy/(1-... (continued fraction). - Paul Barry, Feb 11 2009 T(n,k) = Sum_{j=0..k} binomial(k,j)*T(x,j)*T(y,k-j) for x+y = n. - Dennis P. Walsh, Feb 10 2011 E.g.f (with k fixed): x^k*exp(x). - Dennis P. Walsh, Apr 20 2011 G.f. (with k fixed): k!*x^k/(1-x)^(k+1). - Dennis P. Walsh, Apr 20 2011 For n >= 2 and m >= 2, Sum_{k=0..m-2} S2(n, k+2)*T(m-2, k) = Sum_{p=0..n-2} m^p. S2(n,k) are the Stirling numbers of the second kind A008277. - Tony Foster III, Jul 23 2019 EXAMPLE Triangle begins:   1;   1,  1;   1,  2,  2;   1,  3,  6,   6;   1,  4, 12,  24,   24;   1,  5, 20,  60,  120,   120;   1,  6, 30, 120,  360,   720,    720;   1,  7, 42, 210,  840,  2520,   5040,   5040;   1,  8, 56, 336, 1680,  6720,  20160,  40320,   40320;   1,  9, 72, 504, 3024, 15120,  60480, 181440,  362880,  362880;   1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800;   ... For example, T(4,2)=12 since there are 12 injective functions f:{1,2}->{1,2,3,4}. There are 4 choices for f(1) and then, since f is injective, 3 remaining choices for f(2), giving us 12 ways to construct an injective function. - Dennis P. Walsh, Feb 10 2011 For example, T(5,3)=60 since there are 60 functions f:{1,2,3}->{1,2,3,4,5} with f(x) >= x. There are 5 choices for f(1), 4 choices for f(2), and 3 choices for f(3), giving us 60 ways to construct such a function. - Dennis P. Walsh, Apr 30 2011 MAPLE with(combstruct): for n from 0 to 10 do seq(count(Permutation(n), size=m), m = 0 .. n) od; # Zerinvary Lajos, Dec 16 2007 seq(seq(n!/(n-k)!, k=0..n), n=0..10); # Dennis P. Walsh, Apr 20 2011 seq(print(seq(pochhammer(n-k+1, k), k=0..n)), n=0..6); # Peter Luschny, Mar 26 2015 MATHEMATICA Table[CoefficientList[Series[(1 + x)^m, {x, 0, 20}], x]* Table[n!, {n, 0, m}], {m, 0, 10}] // Grid - Geoffrey Critzer, Mar 16 2010 Table[ Pochhammer[n - k + 1, k], {n, 0, 9}, {k, 0, n}] // Flatten (* or *) Table[ FactorialPower[n, k], {n, 0, 9}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 18 2013, updated Jan 28 2016 *) PROG (PARI) {T(n, k) = if( k<0 || k>n, 0, n!/(n-k)!)}; /* Michael Somos, Nov 14 2002 */ (PARI) {T(n, k) = my(A, p); if( k<0 || k>n, 0, if( n==0, 1, A = matrix(n, n, i, j, x + (i==j)); polcoeff( sum(i=1, n!, if( p = numtoperm(n, i), prod(j=1, n, A[j, p[j]]))), k)))}; /* Michael Somos, Mar 05 2004 */ (Haskell) a008279 n k = a008279_tabl !! n !! k a008279_row n = a008279_tabl !! n a008279_tabl = iterate f  where    f xs = zipWith (+) ( ++ zipWith (*) xs [1..]) (xs ++ ) -- Reinhard Zumkeller, Dec 15 2013, Nov 18 2012 (Sage) for n in range(8): [falling_factorial(n, k) for k in (0..n)] # Peter Luschny, Mar 26 2015 (Magma) /* As triangle */ [[Factorial(n)/Factorial(n-k): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Oct 11 2015 CROSSREFS Row sums give A000522. Cf. A001497, A001498, A136572. T(n,0)=A000012, T(n,1)=A000027, T(n+1,2)=A002378, T(n,3)=A007531, T(n,4)=A052762, and T(n,n)=A000142. Cf. A019538, A090582, A094587, A248727. Sequence in context: A082037 A163649 A110858 * A239572 A056043 A187005 Adjacent sequences:  A008276 A008277 A008278 * A008280 A008281 A008282 KEYWORD nonn,tabl,nice,easy,core AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified September 27 09:43 EDT 2022. Contains 357054 sequences. (Running on oeis4.)