cp's OEIS Frontend

A134991 Triangle of Ward numbers T(n,k) read by rows.

%I A134991 #278 Aug 05 2025 14:39:17
%S A134991 1,1,3,1,10,15,1,25,105,105,1,56,490,1260,945,1,119,1918,9450,17325,
%T A134991 10395,1,246,6825,56980,190575,270270,135135,1,501,22935,302995,
%U A134991 1636635,4099095,4729725,2027025,1,1012,74316,1487200,12122110,47507460,94594500,91891800,34459425
%N A134991 Triangle of Ward numbers T(n,k) read by rows.
%C A134991 This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.
%C A134991 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.
%C A134991 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).
%C A134991 First few polynomials (with a different offset) are
%C A134991   P(0,t) = 0
%C A134991   P(1,t) = 1
%C A134991   P(2,t) = t
%C A134991   P(3,t) = t + 3*t^2
%C A134991   P(4,t) = t + 10*t^2 + 15*t^3
%C A134991   P(5,t) = t + 25*t^2 + 105*t^3 + 105*t^4
%C A134991 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
%C A134991 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
%C A134991 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
%C A134991 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
%C A134991 Named after the American mathematician Morgan Ward (1901-1963). - _Amiram Eldar_, Jun 26 2021
%D A134991 Louis Comtet, Advanced Combinatorics, Reidel, 1974, page 222.
%H A134991 G. C. Greubel, <a href="/A134991/b134991.txt">Rows n = 1..65, flattened</a>
%H A134991 J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, <a href="https://doi.org/10.1016/j.jcta.2014.02.007">Bivariate Generating Functions for a Class of Linear Recurrences, I: General Structure</a>, Journal of Combinatorial Theory, Series A, Vol. 125 (2014), pp. 146-165; <a href="http://arxiv.org/abs/1307.2010">arXiv preprint</a>, arXiv:1307.2010 [math.CO], 2013-2014.
%H A134991 J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.5624">Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications</a>, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
%H A134991 J. Fernando Barbero G., Jesús Salas and Eduardo J. S. Villaseñor, <a href="https://doi.org/10.37236/4814">Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers</a>, Electronic Journal of Combinatorics, Vol. 22, No. 3 (2015), #P3.37.
%H A134991 Andreas Blass, Natasha Dobrinen and Dilip Raghavan, <a href="https://www.jstor.org/stable/43864251">The next best thing to a p-point</a>, The Journal of Symbolic Logic, Vol. 80, No. 3 (2015), pp. 866-900; <a href="http://arxiv.org/abs/1308.3790">arXiv preprint</a>, arXiv:1308.3790 [math.LO], 2013.
%H A134991 David Callan, Toufik Mansour, and Mark Shattuck <a href="https://sciendo.com/es/article/10.1515/puma-2015-0013">Some identities for derangement and Ward number sequences and related bijections</a>, Pure Mathematics and Applications, Vol. 25 (Edition 2), pp. 132-143, 2015.
%H A134991 L. Carlitz, <a href="http://projecteuclid.org/euclid.dmj/1077377459">The coefficients in an asymptotic expansion and certain related numbers</a>, Duke Math. J., Vol. 35, No. 1 (1968), pp. 83-90.
%H A134991 Lane Clark, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00019-9">Asymptotic normality of the Ward numbers</a>, Discrete Math. 203 (1999), no. 1-3, 41-48. [From _N. J. A. Sloane_, Feb 06 2012]
%H A134991 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>.
%H A134991 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 3.
%H A134991 Bishal Deb and Alan D. Sokal, <a href="https://arxiv.org/abs/2507.18959">Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness</a>, arXiv:2507.18959 [math.CO], 2025. See p. 6.
%H A134991 Satyan L. Devadoss, Daoji Huang and Dominic Spadacene, <a href="https://doi.org/10.1137/130947532">Polyhedral covers of tree space</a>, SIAM Journal on Discrete Mathematics, Vol. 28, No. 3 (2014), pp. 1508-1514; <a href="https://arxiv.org/abs/1311.0766">arXiv preprint</a>, arXiv:1311.0766 [math.CO], 2013.
%H A134991 S. Devadoss and J. Morava, <a href="https://arxiv.org/abs/1009.3224">Diagonalizing the genome I: navigation in tree spaces</a>, arXiv:1009.3224v2 [math.AG], 2012.
%H A134991 S. Devadoss and O. Schuh, <a href="https://doi.org/10.1162/leon_a_01475">Cartography of Tree Space</a>, Leonardo (MIT Press Direct), Vol. 53, Issue 3, 2019.
%H A134991 Andreas Dieckmann and Erik Vigren, <a href="https://doi.org/10.3390/sym14061090">A New Result in Form of Finite Triple Sums for a Series from Ramanujan's Notebooks</a>, Symmetry (2022) Vol. 14, No. 6, 1090.
%H A134991 Ming-Jian Ding and Jiang Zeng, <a href="https://arxiv.org/abs/2307.00566">Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions</a>, arXiv:2307.00566 [math.CO], 2023.
%H A134991 Brian Drake, Ira M. Gessel and Guoce Xin, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Gessel/gessel20.html">Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry,</a> J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
%H A134991 Joseph Felsenstein, <a href="https://doi.org/10.2307/2412810">The number of evolutionary trees</a>, Systematic Biology, 27 (1978), pp. 27-33, 1978.
%H A134991 Giovanni Gaiffi, <a href="http://arxiv.org/abs/1404.3395">Nested sets, set partitions and Kirkman-Cayley dissection numbers</a>, arXiv preprint arXiv:1404.3395 [math.CO], 2014.
%H A134991 David M. Jackson, Achim Kempf and Alejandro H. Morales, <a href="https://doi.org/10.1088/1751-8121/aa6abb">A robust generalization of the Legendre transform for QFT</a>, Journal of Physics A: Mathematical and Theoretical, Vol. 50, No. 22 (2017), 225201; <a href="http://arxiv.org/abs/1612.00462">arXiv preprint</a>, arXiv:1612.0046 [hep-th], 2017.
%H A134991 Bradley Robert Jones, <a href="https://www.math.uwaterloo.ca/~kayeats/students/brad_msc.pdf">On tree hook length formulae, Feynman rules and B-series</a>, Master's thesis, Math Dept., Simon Fraser University, 2014. p. 23.
%H A134991 Joachim Kock and Israel Vainsencher, <a href="http://www.math.utah.edu/~yplee/teaching/gw/Koch.pdf">Kontsevich's Formula for Rational Plane Curves</a>, Recife, 1999 (version 31/01/2003).
%H A134991 Toufik Mansour and Mark Shattuck, <a href="https://dx.doi.org/10.1007/s10998-017-0209-9">A polynomial generalization of some associated sequences related to set partitions</a>, Periodica Mathematica Hungarica, Vol. 75, No. 2 (December 2017), pp. 398-412.
%H A134991 MathOverflow, <a href="http://mathoverflow.net/questions/76978/">Face numbers for tropical Grassmannian G'_2,7 simplicial complex?</a>, 2011.
%H A134991 Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%H A134991 Margaret A. Readdy, <a href="https://doi.org/10.1007/s11139-005-4850-1">The pre-WDVV ring of physics and its topology</a>, The Ramanujan Journal, Vol. 10, No. 2 (2005), pp. 269-281; <a href="http://www.ms.uky.edu/~readdy/Papers/pre_WDVV.pdf">preprint</a>, 2002.
%H A134991 Lucas Randazzo, <a href="https://doi.org/10.1007/s11139-019-00185-6">Arboretum for a generalisation of Ramanujan polynomials</a>, The Ramanujan Journal, Vol. 54 (2019), pp. 1-14; <a href="https://arxiv.org/abs/1905.02083">arXiv preprint</a>, arXiv:1905.02083 [math.CO], 2019.
%H A134991 Lucas Randazzo, <a href="https://doi.org/10.1007/s11139-019-00185-6">Combinatoire bijective autour d'arbres et de chemins</a>, doctoral thesis, Université Paris-Est, 2019.
%H A134991 Peter Regner, <a href="http://suuf.cc/phyl-trees/">Phylogenetic Trees: Selected Combinatorial Problems</a>, Master's Thesis, 2012, Institute of Discrete Mathematics and Geometry, TU Vienna, p. 52.
%H A134991 Fuquan Ren, Jean Yeh, and Roberta Rui Zhou, <a href="https://www.mdpi.com/2075-1680/11/6/297/pdf">Context-Free Grammars for Several Triangular Arrays</a>, Axioms, Vol. 11, Issue 6, 297, 2022.
%H A134991 E. G. Santos, <a href="https://arxiv.org/abs/2411.16492">Counting non-attacking chess pieces placements: Bishops and Anassas</a>, arXiv:2411.16492 [math.CO], 2024. See p. 8.
%H A134991 Leonard M. Smiley, <a href="https://arxiv.org/abs/math/0006106">Completion of a Rational Function Sequence of Carlitz</a>, arXiv:math/0006106 [math.CO], 2000.
%H A134991 Pierre Théorêt, <a href="http://www.labmath.uqam.ca/~annales/volumes/19-1/PDF/091-105.pdf">Fonctions génératrices pour une classe d'équations aux différences partielles</a>, Ann. Sci. Math. Quebec 19, 91-105 (1995).
%H A134991 Elena L. Wang and Guoce Xin, <a href="https://arxiv.org/abs/2507.15654">On Ward Numbers and Increasing Schröder Trees</a>, arXiv:2507.15654 [math.CO], 2025. See p. 1.
%H A134991 Morgan Ward, <a href="http://www.jstor.org/stable/2370916">The representations of Stirling's numbers and Stirling's polynomials as sums of factorials</a>, Amer. J. Math., Vol. 56, No. 1 (1934), pp. 87-95.
%H A134991 Bao-Xuan Zhu, <a href="https://arxiv.org/abs/2202.03793">Coefficientwise Hankel-total positivity of row-generating polynomials for the m-Jacobi-Rogers triangle</a>, arXiv:2202.03793 [math.CO], 2022.
%F A134991 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).
%F A134991 From _Tom Copeland_, Oct 26 2008: (Start)
%F A134991 Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
%F A134991 [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).
%F A134991 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).
%F A134991 (End)
%F A134991 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
%F A134991 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
%F A134991 From _Tom Copeland_, Oct 04 2011: (Start)
%F A134991 T(n,k) = (k+1)*T(n-1,k) + (n+k+1)*T(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).
%F A134991 T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
%F A134991 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.
%F A134991 T(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)
%F A134991 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
%F A134991 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
%F A134991 E.g.f.: C(u,t) = (u-t)/(1+t) - W( -((t*exp((u-t)/(1+t)))/(1+t)) ), where W is the principal value of the Lambert W-function. - _Cheng Peng_, Sep 11 2021
%F A134991 The function C(u,t) in the previous formula by Peng is precisely the function A(u,t) given in the initial 2008 formula of this section and the Oct 06 2011 formula from Copeland. As noted in A000169, Euler's tree function is T(x) = -LambertW(-x), where W(x) is the principal branch of Lambert's function, and T(x) is the e.g.f. of A000169. - _Tom Copeland_, May 13 2022
%e A134991 Triangle begins:
%e A134991   1
%e A134991   1   3
%e A134991   1  10   15
%e A134991   1  25  105  105
%e A134991   1  56  490 1260   945
%e A134991   1 119 1918 9450 17325 10395
%e A134991   ...
%t A134991 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 *)
%Y A134991 The same as A269939, with column k = 0 removed.
%Y A134991 A reshaped version of the triangle of associated Stirling numbers of the second kind, A008299.
%Y A134991 A181996 is the mirror image.
%Y A134991 Columns k = 2, 3, 4 are A000247, A000478, A058844.
%Y A134991 Diagonal k = n is A001147.
%Y A134991 Diagonal k = n - 1 is A000457.
%Y A134991 Row sums are A000311.
%Y A134991 Alternating row sums are signed factorials (-1)^(n-1)*A000142(n).
%Y A134991 Cf. A112493.
%K A134991 nonn,tabl
%O A134991 1,3
%A A134991 _Tom Copeland_, Feb 05 2008
%E A134991 Reference to A181996 added by _N. J. A. Sloane_, Apr 05 2012
%E A134991 Further edits by _N. J. A. Sloane_, Jan 24 2020