Search: id:A008292 Results 1-1 of 1 results found. %I A008292 %S A008292 1,1,1,1,4,1,1,11,11,1,1,26,66,26,1,1,57,302,302,57,1,1,120,1191,2416, %T A008292 1191,120,1,1,247,4293,15619,15619,4293,247,1,1,502,14608,88234,156190, %U A008292 88234,14608,502,1,1,1013,47840,455192,1310354,1310354,455192,47840,1013 %N A008292 Triangle of Eulerian numbers T(n,k) read by rows. %C A008292 Coefficients of Eulerian polynomials. Number of permutations of n objects with k-1 rises. Number of increasing rooted trees with n+1 nodes and k leaves. %C A008292 T(n,k)=number of permutations of [n] with k runs. T(n,k)=number of permutations of [n] requiring k readings (see the Knuth reference). T(n,k)=number of permutations of [n] having k distinct entries in its inversion table. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Jun 09 2004 %C A008292 T(n,k) = number of ways to write the Coxeter element s_{e1}s_{e1-e2}s_{e2-e3}s_{e3-e4}...s_{e_{n-1}-e_n} of the reflection group of type B_n, using s_{e_k} and as few reflections of the form s_{e_i+e_j}, where i = 1, 2, ..., n and j is not equal to i, as possible. - Pramook Khungurn (pramook(AT)mit.edu), Jul 07 2004 %C A008292 Subtriangle for k>=1 and n>=1 of triangle A123125 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006 %C A008292 Comment from Stefano Zunino, Oct 25 2006: T(n,k)/n! also represents the n-dimensional volume of the portion of the n-dimensional hyper-cube cut by the (n-1)-dimensional hyperplanes x_1 + x_2 + ... x_n = k, x_1 + x_2 + ... x_n = k-1; or, equivalently, it represents the probability that the sum of n independent random variables with uniform distribution between 0 and 1 is between k-1 and k. %C A008292 [ E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and %C A008292 [ -P(.,t)/(1-t)]^n = n!*Lag[n, E(.,t)/(1-t)] umbrally comprise a combinatorial Laguerre transform pair, where E(n,t) are the Eulerian polynomials and P(n,t) are the polynomials in A131758. - Tom Copeland (tcjpn(AT)msn.com), Sep 30 2007 %C A008292 Comment from Tom Copeland (tcjpn(AT)msn.com), Oct 07 2008: (Start) %C A008292 G(x,t) = 1/ {1 + [1-exp(x t)]/t} = 1 + 1 x + (2 + t) x^2/2! + (6 + 6t + t^2) x^3/3! + ... %C A008292 gives row polynomials for A090582-- reverse f-polynomials for the permutohedra (see A019538). %C A008292 G(x,t-1) = 1 + 1 x + (1 + t) x^2 / 2! + (1 + 4t + t^2) x^3 / 3! + ... %C A008292 gives row polynomials for A008292, the h-polynomials for permutohedra. %C A008292 G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ... %C A008292 gives row polynomials for A028246. %C A008292 (End) %C A008292 A subexceedant function f on [n] is a map f:[n] -> [n] such that 1 <= f(i) <= i for all i, 1 <= i <= n. T(n,k) equals the number of subexceedant functions f of [n] such that the image of f has cardinality k [Mantaci & Rakotondrajao]. Example T(3,2) = 4: if we identify a subexceedant function f with the word f(1)f(2)...f(n) then the subexceedant functions on [3] are 111, 112, 113, 121, 122 and 123 and four of these functions have an image set of cardinality 2. [From Peter Bala (pbala(AT)toucansurf.com), Oct 21 2008] %C A008292 Further to the comments of T. Copeland above, the n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type A_(n-1). The corresponding f-vectors are the rows of A019538. For example, 1 + 4*x + x^2 = y^2 + 6*y + 6 and 1 + 11*x + 11*x^2 + x^3 = y^3 + 14*y^2 + 36*y + 24, where x = y + 1, give [1,6,6] and [1,14,36,24] as the third and fourth rows of A019538. The Hilbert transform of this triangle (see A145905 for the definition) is A047969. See A060187 for the triangle of Eulerian numbers of type B (the h-vectors of the simplicial complexes dual to permutohedra of type B). See A066094 for the array of h-vectors of type D. For tables of restricted Eulerian numbers see A144696 - A144699. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008] %C A008292 For a natural refinement of A008292 with connections to compositional inversion and iterated derivatives, see A145271. [From Tom Copeland (tcjpn(AT)msn.com), Nov 06 2008] %C A008292 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start) %C A008292 The polynomials E(z,n) = numer(sum((-1)^(n+1)*k^n*z^(k-1), k=1..infinity)) for n >=1 lead directly to the triangle of Eulerian numbers. %C A008292 (End) %C A008292 Contribution from Walther Janous (walther.janous(AT)tirol.com), Nov 01 2009: (Start) %C A008292 The (Eulerian) polynomials e(n,x) = SUM(T(n,k+1)*x^k, k = 0 to n-1) turn out to be also as the numerators of the closed-form-expressions of the infinite sums %C A008292 S(p,x):= SUM((j+1)^p*x^j, j = 0 to infinity), that is %C A008292 S(p,x) = e(p,x)/(1-x)^(p+1), whenever |x| < 1 and p is a positive integer. %C A008292 (Please be aware of the inconsistent use of T(n,k) in the part listing the formulae. I adhere tacitly to the first one.) (End) %D A008292 L. Carlitz et al., Permutations and sequences with repetions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351. %D A008292 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243. %D A008292 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260. %D A008292 J. Desarmenien and D. Foata, The signed Eulerian numbers. Discrete Math. 99 (1992), no. 1-3, 49-58. %D A008292 Ehrenborg & Readdy (J Comb. Theory, Series A 81, 121-126). %D A008292 D. Foata, Distributions eule'riennes et mahoniennes sur le group des permutations, pp. 27-49 of M. Aigner, editor, Higher Combinatorics, Reidel, Dordrecht, Holland, 1977. %D A008292 Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1. %D A008292 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254. %D A008292 A. Kerber and K.-J. Thuerlings, Eulerian numbers, Foulkes characters and Lefschetz characters of S_n, Seminaire Lotharingien, Vol. 8 (1984), 31-36. %D A008292 D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, 1998, Vol. 3, p. 47, (exercise 5.1.4 Nr. 20) and p. 605 (solution). %D A008292 P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302. %D A008292 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215. %D A008292 R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996. %D A008292 N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Figure M3416, Academic Press, 1995. %D A008292 H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973, see p. 208. %D A008292 Dominique Foata and Guo-Niu Han, Dimers and new q-tangent numbers, Preprint, 2008. %H A008292 T. D. Noe, Rows 1 to 100 of triangle, flattened. %H A008292 D. Barsky, Analyse p-adique et suites classiques de nombres, Sem. Loth. Comb. B05b (1981) 1-21. %H A008292 J. Desarmenien and D. Foata, The signed Eulerian Numbers %H A008292 E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215. %H A008292 Ira Gessel, The Smith College diploma problem. %H A008292 Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom %H A008292 A. R. Kr\"auter, \"Uber die Permanente gewisser zirkul\"arer Matrizen... %H A008292 A. F. Labossiere, Sobalian Coefficients. %H A008292 A. F. Labossiere, Miscellaneous. %H A008292 R. Mantaci and F. Rakotondrajao, A permutation representation that knows what "Eulerian" means, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001) %H A008292 A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26. %H A008292 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A008292 Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics. %H A008292 L. K. Williams, Enumeration of totally positive Grassmann cells %H A008292 Index entries for sequences related to rooted trees %H A008292 Eric Weisstein's World of Mathematics, Eulerian Number %H A008292 D. Foata, M. Schutzenberger. - Theorie Geometrique des Polynomes Euleriens, Lecture Notes in Math., no.138, Springer Verlag 1970. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008] %H A008292 S. Parker, The Combinatorics of Functional Composition and Inversion, Dissertation, Brandeis Univ. (1993) [From Tom Copeland (tcjpn(AT)msn.com), Nov 10 2008] %F A008292 T(n, k)=k*T(n-1, k)+(n-k+1)*T(n-1, k-1), T(1, 1)=1. T(n, k)=Sum (-1)^j*(k-j)^n*C(n+1, j), j=0..k. %F A008292 E.g.f.: ( e^(zx) - e^x )/( z*e^x - e^(zx) ). %F A008292 For a column listing, n-th term: T(c, n)=c^(n+c-1)+sum(i=1, c-1, (-1)^i/ i!*(c-i)^(n+c-1)*prod(j=1, i, n+c+1-j)) - Randall L. Rathbun (randallr(AT)abac.com), Jan 23 2002 %F A008292 Four characterizations of Eulerian numbers T(i, n) from John Robertson (jpr2718(AT)aol.com), Sep 02, 2002: %F A008292 1. T(0, n)=1 for n>=1, T(i, 1)=0 for i>=1, T(i, n) = (n-i)T(i-1, n-1) + (i+1)T(i, n-1). %F A008292 2. T(i, n) = sum_{j=0}^{i} (-1)^j (n+1 combin j) (i-j+1)^n for n>=1, i>=0. %F A008292 3. Let Cn be the unit cube in R^n with vertices (e_1, e_2, ..., e_n) where each e_i is 0 or 1 and all 2^n combinations are used. Then T(i, n)/n! is the volume of Cn between the hyperplanes x_1 + x_2 + ... + x_n = i and x_1 + x_2 + ... + x_n = i+1. Hence T(i, n)/n! is the probability that i <= X_1 + X_2 + ... + X_n < i+1 where the X_j are independent uniform [0, 1] distributions. - See Ehrenborg & Readdy reference. %F A008292 4. Let f(i, n) = T(i, n)/n!. The f(i, n) are the unique coefficients so that (1/(r-1)^(n+1)) sum_{i=0}^{n-1} f(i, n) r^{i+1} = sum_{j=0}^{infinity} (j^n)/(r^j) whenever n>=1 and abs(r)>1. %F A008292 O.g.f. for n-th row: (1-x)^(n+1)*polylog(-n, x)/x. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 02 2002 %F A008292 Triangle T(n, k), n>0 and k>0, read by rows; given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] (positive integers interspersed with 0's) where DELTA is Deleham's operator defined in A084938. %F A008292 Sum_{k = 1..n} T(n, k)*2^k = A000629(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 05 2004 %F A008292 a(n,k) = Sum_{i=1..n} (-1)^(n-i) * (i^k) * C(k+1,n-i). - Andre F. Labossiere (boronali(AT)laposte.net), Aug 16 2006 %F A008292 Formulae and comments from Tom Copeland, Oct 10 2007 (Start): Bell(n, x) = sum(j=0,...,n) S2(n,j) * x^j = sum(j=0,...,n) E(n,j) * Lag(n, -x, j-n) = sum(j=0,...,n) [ E(n,j)/n! ] * [ n!*Lag(n,-x, j-n) ] = sum(j=0,...,n) E(n,j) * C(Bell(.,x)+j,n) umbrally where Bell(n,x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; Lag(n,x, m), the associated Laguerre polynomials of order m; and C(x,y) = x!/[ y!*(x-y)! ]. %F A008292 For x = 0, the equation gives sum(j=0,...,n) E(n,j) * C(j,n) = 1 for n = 0 and 0 for all other n. By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*C(y,n), for x in the equation, the Worpitzky identity is obtained; y^n = sum(j=0, ...,n) E(n,j) * C(y+j,n) . %F A008292 Note that E(n,j)/n! = E(n,j)/ {sum(k=0,..,n)E(n,k)}. Also [n!*Lag(n,-1, j-n)] is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to the equation for x=1; n!*Bell(n,1) = n!*sum(j=0,...,n) S2(n,j) = sum(j=0,...,n) {E(n,j) * [n!*Lag(n,-1, j-n)]} . (End) %F A008292 From the relations between the h- and f-polynomials of permutohedra and reciprocals of e.g.f.s described in A049019: (t-1)[(t-1)d/dx]^n 1/ (t-exp(x)) evaluated at x=0 gives the n-th Eulerian row polynomial in t and the n-th row polynomial in (t-1) of A019538 and A090582. From the Comtet and Copeland references in A139605: [(t+exp(x)-1)d/ dx]^(n+1) x gives pairs of the Eulerian polynomials in t as the coefficients of x^0 and x^1 in its Taylor series expansion in x. [From Tom Copeland (tcjpn(AT)msn.com), Oct 05 2008] %e A008292 Triangle begins: %e A008292 1; %e A008292 1,1; %e A008292 1,4,1; %e A008292 1,11,11,1; %e A008292 1,26,66,26,1; %e A008292 1,57,302,302,57,1; %e A008292 ... %o A008292 (PARI) T(n,k)=if(k<1|k>n,0, if(n==1,1,k*T(n-1,k)+(n-k+1)*T(n-1,k-1))) %o A008292 (PARI) {T(n,k)=sum(j=0,k,(-1)^j*(k-j)^n*binomial(n+1,j))} {A008292(c, n)=c^(n+c-1)+sum(i=1,c-1,(-1)^i/i!*(c-i)^(n+c-1)*prod(j=1,i,n+c+1-j))} %Y A008292 Cf. A014630, A030196, A055325. Row sums give A000142(n). %Y A008292 Cf. A027656 A084938, A049061, A008275, A008277, A087127. %Y A008292 Columns 2 through 8: A000295, A000460, A000498, A000505, A000514, A001243, A001244. %Y A008292 Cf. A062253, noting that A008292 gives the (unsigned) polynomial coefficients of (kn). Also note that taking alternating differences of rows with an odd number of terms (e.g. 1=1, -1+4-1=2, 1-26+66-26+1=16, -1+120-1191+2416-1191+120-1=272 etc.) gives A000182. - Henry Bottomley (se16(AT)btinternet.com), Jun 15 2001 %Y A008292 A019538 (f-vectors type A permutohedra), A047969 (Hilbert transform), A060187 (h-vectors type B permutohedra), A066094. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008] %Y A008292 Sequence in context: A154986 A154983 A156534 this_sequence A157221 A146967 A156049 %Y A008292 Adjacent sequences: A008289 A008290 A008291 this_sequence A008293 A008294 A008295 %K A008292 nonn,tabl,nice,eigen,core,new %O A008292 1,5 %A A008292 N. J. A. Sloane (njas(AT)research.att.com). %E A008292 Thanks to Michael Somos for additional comments. Further comments from Christian G. Bower (bowerc(AT)usa.net), May 12 2000 Search completed in 0.003 seconds