1,5

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.

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

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

Subtriangle for k>=1 and n>=1 of triangle A123125 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006

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.

[ E(.,t)/(1-t)]^n = n!*Lag[n,-P(.,t)/(1-t)] and

[ -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

Comment from Tom Copeland (tcjpn(AT)msn.com), Oct 07 2008: (Start)

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! + ...

gives row polynomials for A090582-- reverse f-polynomials for the permutohedra (see A019538).

G(x,t-1) = 1 + 1 x + (1 + t) x^2 / 2! + (1 + 4t + t^2) x^3 / 3! + ...

gives row polynomials for A008292, the h-polynomials for permutohedra.

G[(t+1)x,-1/(t+1)] = 1 + (1+ t) x + (1 + 3t + 2 t^2) x^2 / 2! + ...

gives row polynomials for A028246.

(End)

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]

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]

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]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 24 2009: (Start)

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.

(End)

Contribution from Walther Janous (walther.janous(AT)tirol.com), Nov 01 2009: (Start)

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

S(p,x):= SUM((j+1)^p*x^j, j = 0 to infinity), that is

S(p,x) = e(p,x)/(1-x)^(p+1), whenever |x| < 1 and p is a positive integer.

(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)

L. Carlitz et al., Permutations and sequences with repetions by number of increases, J. Combin. Theory, 1 (1966), 350-374, p. 351.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260.

J. Desarmenien and D. Foata, The signed Eulerian numbers. Discrete Math. 99 (1992), no. 1-3, 49-58.

Ehrenborg & Readdy (J Comb. Theory, Series A 81, 121-126).

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.

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.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.

A. Kerber and K.-J. Thuerlings, Eulerian numbers, Foulkes characters and Lefschetz characters of S_n, Seminaire Lotharingien, Vol. 8 (1984), 31-36.

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).

P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

N. J. A. Sloane and S. Plouffe, The Encyclopedia of Integer Sequences, Figure M3416, Academic Press, 1995.

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, 1973, see p. 208.

Dominique Foata and Guo-Niu Han, Dimers and new q-tangent numbers, Preprint, 2008.

T. D. Noe, Rows 1 to 100 of triangle, flattened.

D. Barsky, Analyse p-adique et suites classiques de nombres, Sem. Loth. Comb. B05b (1981) 1-21.

J. Desarmenien and D. Foata, The signed Eulerian Numbers

E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.

Ira Gessel, The Smith College diploma problem.

Matthew Hubbard and Tom Roby, Pascal's Triangle From Top to Bottom

A. R. Kr\"auter, \"Uber die Permanente gewisser zirkul\"arer Matrizen...

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

R. Mantaci and F. Rakotondrajao, A permutation representation that knows what "Eulerian" means, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001)

A. Randrianarivony and J. Zeng, Une famille des polynomes qui interpole plusieurs suites..., Adv. Appl. Math. 17 (1996), 1-26.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

L. K. Williams, Enumeration of totally positive Grassmann cells

Index entries for sequences related to rooted trees

Eric Weisstein's World of Mathematics, Eulerian Number

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]

S. Parker, The Combinatorics of Functional Composition and Inversion, Dissertation, Brandeis Univ. (1993) [From Tom Copeland (tcjpn(AT)msn.com), Nov 10 2008]

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.

E.g.f.: ( e^(zx) - e^x )/( z*e^x - e^(zx) ).

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

Four characterizations of Eulerian numbers T(i, n) from John Robertson (jpr2718(AT)aol.com), Sep 02, 2002:

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).

2. T(i, n) = sum_{j=0}^{i} (-1)^j (n+1 combin j) (i-j+1)^n for n>=1, i>=0.

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.

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.

O.g.f. for n-th row: (1-x)^(n+1)*polylog(-n, x)/x. - Vladeta Jovovic (vladeta(AT)eunet.rs), Sep 02 2002

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.

Sum_{k = 1..n} T(n, k)*2^k = A000629(n) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jun 05 2004

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

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)! ].

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) .

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)

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]

Triangle begins:

1;

1,1;

1,4,1;

1,11,11,1;

1,26,66,26,1;

1,57,302,302,57,1;

...

(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)))

(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))}

Cf. A014630, A030196, A055325. Row sums give A000142(n).

Cf. A027656 A084938, A049061, A008275, A008277, A087127.

Columns 2 through 8: A000295, A000460, A000498, A000505, A000514, A001243, A001244.

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

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]

Adjacent sequences: A008289 A008290 A008291 this_sequence A008293 A008294 A008295

Sequence in context: A154986 A154983 A156534 this_sequence A157221 A146967 A156049

nonn,tabl,nice,eigen,core,new

N. J. A. Sloane (njas(AT)research.att.com).

Thanks to Michael Somos for additional comments. Further comments from Christian G. Bower (bowerc(AT)usa.net), May 12 2000