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A134991 Triangle of Ward numbers T(n,k) read by rows. 20
1, 1, 3, 1, 10, 15, 1, 25, 105, 105, 1, 56, 490, 1260, 945, 1, 119, 1918, 9450, 17325, 10395, 1, 246, 6825, 56980, 190575, 270270, 135135, 1, 501, 22935, 302995, 1636635, 4099095, 4729725, 2027025, 1, 1012, 74316, 1487200, 12122110, 47507460 (list; table; graph; refs; listen; history; text; internal format)



This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.

This is also a row-reversed version of A181996 (with an additional leading 1) - see the table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table.

The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).

First few polynomials (with a different offset) are

P(0,t) = 0

P(1,t) = 1

P(2,t) = t

P(3,t) = t + 3 t^2

P(4,t) = t + 10 t^2 + 15 t^3

P(5,t) = t + 25 t^2 + 105 t^3 + 105 t^4

These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011

A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n , for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 =0. - Tom Copeland, Oct 08 2011

Beginning with the second column, the rows give the faces of the Whitehouse simplicial complex with the fourth order complex being three isolated vertices and the fifth order being the Petersen graph with 10 vertices and 15 edges (cf. Readdy). - Tom Copeland, Oct 03 2014

Stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. Cf. pages 20 and 30 of the Kock and Vainsencher reference and references in A134685. - Tom Copeland, May 18 2017


L. Comtet, Advanced Combinatorics, Reidel, 1974, page 222.

J. Kock and I. Vainsencher, Kontsevich's Formula for Rational Plane Curves, Recife, 1999 (version 31/01/2003).


G. C. Greubel, Rows n = 1..65, 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.

J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.

J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.

A. Blass, N. Dobrinen, D. Raghavan, The next best thing to a P-point, arXiv preprint arXiv:1308.3790 [math.LO], 2013.

L. Carlitz, The coefficients in an asymptotic expansion and certain related numbers, Duke Math. J., vol 35 (1968), p. 83-90.

Lane Clark, Asymptotic normality of the Ward numbers, Discrete Math. 203 (1999), no. 1-3, 41-48. [From N. J. A. Sloane, Feb 06 2012]

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

S. Devadoss, D. Huang, D. Spadacene, Polyhedral covers of tree space, arXiv preprint arXiv:1311.0766 [math.CO], 2013.

Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.

G. Gaiffi, Nested sets, set partitions and Kirkman-Cayley dissection numbers, arXiv preprint arXiv:1404.3395 [math.CO], 2014.

D. Jackson, A. Kempf, A. Morales, A robust generalization of the Legendre transform for QFT, arXiv:1612.0046 [hep-th], 2017.

B. Jones, On tree hook length formulae, Feynman rules and B-series, Master's thesis, Math Dept., Simon Fraser University, 2014. p. 23.

Toufik Mansour and Mark Shattuck, A polynomial generalization of some associated sequences related to set partitions, Periodica Mathematica Hungarica, December 2017, Volume 75, Issue 2, pp. 398-412.

MathOverflow, Face numbers for tropical Grassmannian G'_2,7 simplicial complex?

Andrew Elvey Price, Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

M. A. Readdy, The pre-WDVV ring of physics and its topology, preprint, 2002.

Lucas Randazzo, Arboretum for a generalization of Ramanujan polynomials, arXiv:1905.02083 [math.CO], 2019.

P. Regner, Phylogenetic Trees: Selected Combinatorial Problems, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, p. 52.

L. M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.

M. Ward, The representations of Stirling's numbers and Stirling's polynomials as sums of factorials, Amer. J. Math., 56 (1934), pp. 87-95.


E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for A000311. These results are a special case of A134685 with u(x) = B(x).

From Tom Copeland, Oct 26 2008: (Start)

Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),

[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).

Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) = A000311(m).


Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of A019538, then the n-th row polynomial in t of the table A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). - Tom Copeland, Sep 05 2011

The polynomials (1+t)/t*P(n,t) are the row polynomials of A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. - Peter Bala, Sep 30 2011

From Tom Copeland, Oct 04 2011: (Start)

a(n,k) = (k+1)a(n-1,k)+(n+k+1)a(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow). a(n,k)= k a(n-1,k)+(n+k-1)a(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).

P(n,t) = (1/(1+t))^n Sum_{k>=1} k^(n+k-1) [(u*exp(-u)]^k / k! with u=(t/(t+1)) for n>1; therefore, Sum_{k>=1} (-1)^k k^(n+k-1) x^k/k!

  = [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.

a(n,k) = Sum_{i=0..k} ((-1)^i binomial(n+k,i) Sum_{j=0..k-i} (-1)^j (k-i-j)^(n+k-i)/(j!(k-i-j)!)) from relation to A008299. (End)

The e.g.f. A(x,t) = -v * ( Sum_{j=>1} D(j-1,u) (-z)^j / j! ) where u = (x-t)/(1+t), v = 1+u, z = x/((1+t) v^2) and D(j-1,u) are the polynomials of A042977. dA/dx = 1/((1+t)(v-A)) = 1/(1-t*(exp(A)-1)). - Tom Copeland, Oct 06 2011

The general results on the convolution of the refined partition polynomials of A134685, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016


t[n_, k_] := Sum[(-1)^i*Binomial[n, i]*Sum[(-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!), {j, 0, k-i}], {i, 0, k}]; row[n_] := Table[t[k, k-n], {k, n+1, 2*n}]; Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Apr 23 2014, after A008299 *)


Essentially the same as the triangle of associated Stirling numbers of the second kind, A008299.

A181996 is the mirror image.

Cf. A000457, A001147, A112493, A269939.

Sequence in context: A068438 A064060 A176740 * A212930 A225725 A095327

Adjacent sequences:  A134988 A134989 A134990 * A134992 A134993 A134994




Tom Copeland, Feb 05 2008


Reference to A181996 added by N. J. A. Sloane, Apr 05 2012. Further edits by N. J. A. Sloane, Jan 24 2020.



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Last modified March 2 17:48 EST 2021. Contains 341754 sequences. (Running on oeis4.)