This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A134264 #383 Mar 30 2024 23:09:14
%S A134264 1,1,1,1,1,3,1,1,4,2,6,1,1,5,5,10,10,10,1,1,6,6,3,15,30,5,20,30,15,1,
%T A134264 1,7,7,7,21,42,21,21,35,105,35,35,70,21,1,1,8,8,8,4,28,56,56,28,28,56,
%U A134264 168,84,168,14,70,280,140,56,140,28,1,1,9,9,9,9,36,72
%N A134264 Coefficients T(j, k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions and primitive parking functions. T(n,k) for n >= 1 and 1 <= k <= A000041(n-1), an irregular triangle read by rows.
%C A134264 Coefficients are listed in Abramowitz and Stegun order (A036036).
%C A134264 Given an invertible function f(t) analytic about t=0 (or a formal power series) with f(0)=0 and Df(0) not equal to 0, form h(t) = t / f(t) and let h_n denote the coefficient of t^n in h(t).
%C A134264 Lagrange inversion gives the compositional inverse about t=0 as g(t) = Sum_{j>=1} ( t^j * (1/j) * Sum_{permutations s with s(1) + s(2) + ... + s(j) = j - 1} h_s(1) * h_s(2) * ... * h_s(j) ) = t * T(1,1) * h_0 + Sum_{j>=2} ( t^j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) ), where H(j-1,k ; h_0,h_1,...) is the k-th partition for h_1 through h_(j-1) corresponding to n=j-1 on page 831 of Abramowitz and Stegun (ordered as in A&S) with (h_0)^(j-m)=(h_0)^(n+1-m) appended to each partition subsumed under n and m of A&S.
%C A134264 Denoting h_n by (n') for brevity, to 8th order in t,
%C A134264 g(t) = t * (0')
%C A134264 + t^2 * [ (0') (1') ]
%C A134264 + t^3 * [ (0')^2 (2') + (0') (1')^2 ]
%C A134264 + t^4 * [ (0')^3 (3') + 3 (0')^2 (1') (2') + (0') (1')^3 ]
%C A134264 + t^5 * [ (0')^4 (4') + 4 (0')^3 (1') (3') + 2 (0')^3 (2')^2 + 6 (0')^2 (1')^2 (2') + (0') (1')^4 ]
%C A134264 + t^6 * [ (0')^5 (5') + 5 (0')^4 (1') (4') + 5 (0')^4 (2') (3') + 10 (0')^3 (1')^2 (3') + 10 (0')^3 (1') (2')^2 + 10 (0')^2 (1')^3 (2') + (0') (1')^5 ]
%C A134264 + t^7 * [ (0')^6 (6') + 6 (0')^5 (1') (5') + 6 (0')^5 (2') (4') + 3 (0')^5 (3')^2 + 15 (0')^4 (1')^2 (4') + 30 (0')^4 (1') (2') (3') + 5 (0')^4 (2')^3 + 20 (0')^3 (1')^3 (3') + 30 (0')^3 (1')^2 (2')^2 + 15 (0')^2 (1')^4 (2') + (0') (1')^6]
%C A134264 + t^8 * [ (0')^7 (7') + 7 (0')^6 (1') (6') + 7 (0')^6 (2') (5') + 7 (0')^6 (3') (4') + 21 (0')^5 (1')^2* (5') + 42 (0')^5 (1') (2') (4') + 21 (0')^5 (1') (3')^2 + 21 (0')^5 (2')^2 (3') + 35 (0')^4 (1')^3 (4') + 105 (0)^4 (1')^2 (2') (3') + 35 (0')^4 (1') (2')^3 + 35 (0')^3 (1')^4 (3') + 70 (0')^3 (1')^3 (2')^2 + 21 (0')^2 (1')^5 (2') + (0') (1')^7 ]
%C A134264 + ..., where from the formula section, for example, T(8,1',2',...,7') = 7! / ((8 - (1'+ 2' + ... + 7'))! * 1'! * 2'! * ... * 7'!) are the coefficients of the integer partitions (1')^1' (2')^2' ... (7')^7' in the t^8 term.
%C A134264 A125181 is an extended, reordered version of the above sequence, omitting the leading 1, with alternate interpretations.
%C A134264 If the coefficients of partitions with the same exponent for h_0 are summed within rows, A001263 is obtained, omitting the leading 1.
%C A134264 From identification of the elements of the inversion with those on page 25 of the Ardila et al. link, the coefficients of the irregular table enumerate non-crossing partitions on [n]. - _Tom Copeland_, Oct 13 2014
%C A134264 From _Tom Copeland_, Oct 28-29 2014: (Start)
%C A134264 Operating with d/d(1') = d/d(h_1) on the n-th partition polynomial Prt(n;h_0,h_1,..,h_n) in square brackets above associated with t^(n+1) generates n * Prt(n-1;h_0,h_1,..,h_(n-1)); therefore, the polynomials are an Appell sequence of polynomials in the indeterminate h_1 when h_0=1 (a special type of Sheffer sequence).
%C A134264 Consequently, umbrally, [Prt(.;1,x,h_2,..) + y]^n = Prt(n;1,x+y,h_2,..); that is, Sum_{k=0..n} binomial(n,k) * Prt(k;1,x,h_2,..) * y^(n-k) = Prt(n;1,x+y,h_2,..).
%C A134264 Or, e^(x*z) * exp[Prt(.;1,0,h_2,..) * z] = exp[Prt(.;1,x,h_2,..) * z]. Then with x = h_1 = -(1/2) * d^2[f(t)]/dt^2 evaluated at t=0, the formal Laplace transform from z to 1/t of this expression generates g(t), the comp. inverse of f(t), when h_0 = 1 = df(t)/dt eval. at t=0.
%C A134264 I.e., t / (1 - t*(x + Prt(.;1,0,h_2,..))) = t / (1 - t*Prt(.;1,x,h_2,..)) = g(t), interpreted umbrally, when h_0 = 1.
%C A134264 (End)
%C A134264 Connections to and between arrays associated to the Catalan (A000108 and A007317), Riordan (A005043), Fibonacci (A000045), and Fine (A000957) numbers and to lattice paths, e.g., the Motzkin, Dyck, and Łukasiewicz, can be made explicit by considering the inverse in x of the o.g.f. of A104597(x,-t), i.e., f(x) = P(Cinv(x),t-1) = Cinv(x) / (1 + (t-1)*Cinv(x)) = x*(1-x) / (1 + (t-1)*x*(1-x)) = (x-x^2) / (1 + (t-1)*(x-x^2)), where Cinv(x) = x*(1-x) is the inverse of C(x) = (1 - sqrt(1-4*x)) / 2, a shifted o.g.f. for the Catalan numbers, and P(x,t) = x / (1+t*x) with inverse Pinv(x,t) = -P(-x,t) = x / (1-t*x). Then h(x,t) = x / f(x,t) = x * (1+(t-1)Cinv(x)) / Cinv(x) = 1 + t*x + x^2 + x^3 + ..., i.e., h_1=t and all other coefficients are 1, so the inverse of f(x,t) in x, which is explicitly in closed form finv(x,t) = C(Pinv(x,t-1)), is given by A091867, whose coefficients are sums of the refined Narayana numbers above obtained by setting h_1=(1')=t in the partition polynomials and all other coefficients to one. The group generators C(x) and P(x,t) and their inverses allow associations to be easily made between these classic number arrays. - _Tom Copeland_, Nov 03 2014
%C A134264 From _Tom Copeland_, Nov 10 2014: (Start)
%C A134264 Inverting in x with t a parameter, let F(x;t,n) = x - t*x^(n+1). Then h(x) = x / F(x;t,n) = 1 / (1-t*x^n) = 1 + t*x^n + t^2*x^(2n) + t^3*x^(3n) + ..., so h_k vanishes unless k = m*n with m an integer in which case h_k = t^m.
%C A134264 Finv(x;t,n) = Sum_{j>=0} {binomial((n+1)*j,j) / (n*j + 1)} * t^j * x^(n*j + 1), which gives the Catalan numbers for n=1, and the Fuss-Catalan sequences for n>1 (see A001764, n=2). [Added braces to disambiguate the formula. - _N. J. A. Sloane_, Oct 20 2015]
%C A134264 This relation reveals properties of the partitions and sums of the coefficients of the array. For n=1, h_k = t^k for all k, implying that the row sums are the Catalan numbers. For n = 2, h_k for k odd vanishes, implying that there are no blocks with only even-indexed h_k on the even-numbered rows and that only the blocks containing only even-sized bins contribute to the odd-row sums giving the Fuss-Catalan numbers for n=2. And so on, for n > 2.
%C A134264 These relations are reflected in any combinatorial structures enumerated by this array and the partitions, such as the noncrossing partitions depicted for a five-element set (a pentagon) in Wikipedia.
%C A134264 (End)
%C A134264 From _Tom Copeland_, Nov 12 2014: (Start)
%C A134264 An Appell sequence possesses an umbral inverse sequence (cf. A249548). The partition polynomials here, Prt(n;1,h_1,...), are an Appell sequence in the indeterminate h_1=u, so have an e.g.f. exp[Prt(.;1,u,h_2...)*t] = e^(u*t) * exp[Prt(.;1,0,h2,...)*t] with umbral inverses with an e.g.f e^(-u*t) / exp[Prt(.;1,0,h2,...)*t]. This makes contact with the formalism of A133314 (cf. also A049019 and A019538) and the signed, refined face partition polynomials of the permutahedra (or their duals), which determine the reciprocal of exp[Prt(.,0,u,h2...)*t] (cf. A249548) or exp[Prt(.;1,u,h2,...)*t], forming connections among the combinatorics of permutahedra and the noncrossing partitions, Dyck paths and trees (cf. A125181), and many other important structures isomorphic to the partitions of this entry, as well as to formal cumulants through A127671 and algebraic structures of Lie algebras. (Cf. relationship of permutahedra with the Eulerians A008292.)
%C A134264 (End)
%C A134264 From _Tom Copeland_, Nov 24 2014: (Start)
%C A134264 The n-th row multiplied by n gives the number of terms in the homogeneous symmetric monomials generated by [x(1) + x(2) + ... + x(n+1)]^n under the umbral mapping x(m)^j = h_j, for any m. E.g., [a + b + c]^2 = [a^2 + b^2 + c^2] + 2 * [a*b + a*c + b*c] is mapped to [3 * h_2] + 2 * [3 * h_1^2], and 3 * A134264(3) = 3 *(1,1)= (3,3) the number of summands in the two homogeneous polynomials in the square brackets. For n=3, [a + b + c + d]^3 = [a^3 + b^3 + ...] + 3 [a*b^2 + a*c^2 + ...] + 6 [a*b*c + a*c*d + ...] maps to [4 * h_3] + 3 [12 * h_1 * h_2] + 6 [4 * (h_1)^3], and the number of terms in the brackets is given by 4 * A134264(4) = 4 * (1,3,1) = (4,12,4).
%C A134264 The further reduced expression is 4 h_3 + 36 h_1 h_2 + 24 (h_1)^3 = A248120(4) with h_0 = 1. The general relation is n * A134264(n) = A248120(n) / A036038(n-1) where the arithmetic is performed on the coefficients of matching partitions in each row n.
%C A134264 Abramowitz and Stegun give combinatorial interpretations of A036038 and relations to other number arrays.
%C A134264 This can also be related to repeated umbral composition of Appell sequences and topology with the Bernoulli numbers playing a special role. See the Todd class link.
%C A134264 (End)
%C A134264 These partition polynomials are dubbed the Voiculescu polynomials on page 11 of the He and Jejjala link. - _Tom Copeland_, Jan 16 2015
%C A134264 See page 5 of the Josuat-Verges et al. reference for a refinement of these partition polynomials into a noncommutative version composed of nondecreasing parking functions. - _Tom Copeland_, Oct 05 2016
%C A134264 (Per Copeland's Oct 13 2014 comment.) The number of non-crossing set partitions whose block sizes are the parts of the n-th integer partition, where the ordering of integer partitions is first by total, then by length, then lexicographically by the reversed sequence of parts. - _Gus Wiseman_, Feb 15 2019
%C A134264 With h_0 = 1 and the other h_n replaced by suitably signed partition polynomials of A263633, the refined face partition polynomials for the associahedra of normalized A133437 with a shift in indices are obtained (cf. In the Realm of Shadows). - _Tom Copeland_, Sep 09 2019
%C A134264 Number of primitive parking functions associated to each partition of n. See Lemma 3.8 on p. 28 of Rattan. - _Tom Copeland_, Sep 10 2019
%C A134264 With h_n = n + 1, the d_k (A006013) of Table 2, p. 18, of Jong et al. are obtained, counting the n-point correlation functions in a quantum field theory. - _Tom Copeland_, Dec 25 2019
%C A134264 By inspection of the diagrams on Robert Dickau's website, one can see the relationship between the monomials of this entry and the connectivity of the line segments of the noncrossing partitions. - _Tom Copeland_, Dec 25 2019
%C A134264 Speicher has examples of the first four inversion partition polynomials on pp. 22 and 23 with his k_n equivalent to h_n = (n') here with h_0 = 1. Identifying z = t, C(z) = t/f(t) = h(t), and M(z) = f^(-1)(t)/t, then statement (3), on p. 43, of Theorem 3.26, C(z M(z)) = M(z), is equivalent to substituting f^(-1)(t) for t in t/f(t), and statement (4), M(z/C(z)) = C(z), to substituting f(t) for t in f^(-1)(t)/t. - _Tom Copeland_, Dec 08 2021
%C A134264 Given a Laurent series of the form f(z) = 1/z + h_1 + h_2 z + h_3 z^2 + ..., the compositional inverse is f^(-1)(z) = 1/z + Prt(1;1,h_1)/z^2 + Prt(2;1,h_1,h_2)/z^3 + ... = 1/z + h_1/z^2 + (h_1^2 + h_2)/z^3 + (h_1^3 + 3 h_1 h_2 + h_3)/z^4 + (h_1^4 + 6 h_1^2 h_2 + 4 h_1 h_3 + 2 h_2^2 + h_4)/z^5 + ... for which the polynomials in the numerators are the partition polynomials of this entry. For example, this formula applied to the q-expansion of Klein's j-invariant / function with coefficients A000521, related to monstrous moonshine, gives the compositional inverse with the coefficients A091406 (see He and Jejjala). - _Tom Copeland_, Dec 18 2021
%C A134264 The partition polynomials of A350499 'invert' the polynomials of this entry giving the indeterminates h_n. A multinomial formula for the coefficients of the partition polynomials of this entry, equivalent to the multinomial formula presented in the first four sentences of the formula section below, is presented in the MathOverflow question referenced in A350499. - _Tom Copeland_, Feb 19 2022
%D A134264 A. Nica and R. Speicher (editors), Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series: 335, Cambridge University Press, 2006 (see in particular, Eqn. 9.14 on p. 141, enumerating noncrossing partitions).
%H A134264 Andrew Howroyd, <a href="/A134264/b134264.txt">Table of n, a(n) for n = 1..2087</a> (rows 1..20)
%H A134264 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A134264 A. Alexandrov, <a href="http://arxiv.org/abs/1404.3402">Enumerative geometry, tau-functions, and Heisenberg-Virasoro algebra</a>, arXiv:hep-th/1404.3402v3, 2015 (p.22).
%H A134264 M. Ashelevich and W. Mlotkowski, <a href="https://arxiv.org/abs/1001.1540">Semigroups of distributions with linear Jacobi parameters</a>, arxiv.org/abs/1001.1540v3 [math.CO], p. 9, 2011.
%H A134264 F. Ardila, F. Rincon, and L. Williams, <a href="http://arxiv.org/abs/1308.2698">Positroids and non-crossing partitions</a>, arXiv preprint arXiv:1308.2698v2 [math.CO], 2013 (p. 25).
%H A134264 T. Banica, S. Belinschi, M. Capitaine, and B. Collins, <a href="https://arxiv.org/abs/0710.5931">Free Bessel laws</a>, arXiv preprint arXiv:0710.5931 [math.PR], 2008.
%H A134264 Freddy Cachazo and Bruno Giménez Umbert, <a href="https://arxiv.org/abs/2205.02722">Connecting Scalar Amplitudes using The Positive Tropical Grassmannian</a>, arXiv:2205.02722 [hep-th], 2022.
%H A134264 David Callan, <a href="http://arxiv.org/abs/0711.4841">Sets, Lists and Noncrossing Partitions </a>, arXiv:0711.4841 [math.CO], 2008.
%H A134264 B. Collins, I. Nechita, and K. Zyczkowski, <a href="https://arxiv.org/abs/1003.3075">Random graph states, maximal flow and Fuss-Catalan distributions</a>, arXiv preprint arXiv:1003.3075 [quant-ph], 2010.
%H A134264 Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/11/02/compositional-inversion-and-appell-sequences/">Compositional inversion and Appell sequences</a>, Nov 2, 2014.
%H A134264 Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/14/the-hirzebruch-criterion-fo-the-todd-class/">The Hirzebruch criterion for the Todd class</a> Dec. 14, 2014.
%H A134264 Tom Copeland, <a href="http://tcjpn.wordpress.com/2014/12/23/appell-ops-cumulants-noncrossing-partitions-dyck-lattice-paths-and-inversion/">Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion</a> Dec. 23, 2014.
%H A134264 Tom Copeland, <a href="https://tcjpn.wordpress.com/2018/01/23/formal-group-laws-and-binomial-sheffer-sequences/">Formal group laws and binomial Sheffer sequences</a>, 2018.
%H A134264 Tom Copeland, <a href="https://tcjpn.wordpress.com/2019/09/13/associahedra-noncrossing-partitions-and-an-umbral-algebra-of-power-series/">In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms</a>, 2019.
%H A134264 Tom Copeland, <a href="https://tcjpn.wordpress.com/2022/01/29/a-taste-of-moonshine-in-free-moments/">A Taste of Moonshine in Free Moments</a>, 2022.
%H A134264 R. Dickau, <a href="https://www.robertdickau.com/noncrossingpartitions.html">Non-crossing partitions</a>, Robert Dickau's website, 2012.
%H A134264 K. Ebrahimi-Fard, L. Foissy, J. Kock, and F. Patras, <a href="https://arxiv.org/abs/1907.01190">Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations</a>, arXiv:1907.01190 [math.CO], 2019, p. 25.
%H A134264 K. Ebrahimi-Fard and F. Patras, <a href="https://arxiv.org/abs/1409.5664">Cumulants, free cumulants and half-shuffles</a>, arXiv:1409.5664v2 [math.CO], 2015, p. 12.
%H A134264 K. Ebrahimi-Fard and F. Patras, <a href="https://arxiv.org/abs/1502.02748">The splitting process in free probability theory</a>, arXiv:1502.02748 [math.CO], 2015, p. 3.
%H A134264 Y. He and V. Jejjala, <a href="http://arxiv.org/abs/hep-th/0307293">Modular Matrix Models</a>, arXiv:hep-th/0307293, 2003.
%H A134264 J. Jong, A. Hock, and R. Wulkenhaar <a href="https://arxiv.org/abs/1904.11231">Catalan tables and a recursion relation in noncommutative quantum field theory</a>, arXiv preprint arXiv:1904.11231 [math-ph], 2019.
%H A134264 M. Josuat-Verges, F. Menous, J. Novelli, and J. Thibon, <a href="http://arxiv.org/abs/1604.04759">Noncommutative free cumulants</a>, arxiv.org/abs/1604.04759 [math.CO], 2016.
%H A134264 Germain Kreweras, <a href="https://doi.org/10.1016/0012-365X(72)90041-6">Sur les partitions non croisées d'un cycle</a>, Discrete Math. 1 333-350 (1972).
%H A134264 C. Lenart, <a href="https://doi.org/10.1016/0012-365X(72)90041-6">Lagrange inversion and Schur functions</a>, Journal of Algebraic Combinatorics, Vol. 11, Issue 1, p. 69-78, 2000, (see p. 70, Eqn. 1.2).
%H A134264 M. Mastnak and A. Nica, <a href="https://arxiv.org/abs/0807.4169">Hopf algebras and the logarithm of the S-transform in free probability</a>, arXiv:0807.4169v2 [math.OA], p. 28, 2008-2009.
%H A134264 MathOverflow, <a href="https://mathoverflow.net/questions/338905/guises-of-the-noncrossing-partitions-ncps">Guises of the Noncrossing Partitions (NCPs)</a>, an MO question posed by Tom Copeland, 2019.
%H A134264 MathOverflow, <a href="https://mathoverflow.net/questions/412573/combinatorial-diagrammatic-models-for-free-moment-partition-polynomials">Combinatorial/diagrammatic models for free moment partition polynomials</a>, an MO question posed by Tom Copeland, 2021.
%H A134264 J. McCammond, <a href="http://www.math.ucsb.edu/~jon.mccammond/papers/nc-survey-official.pdf">Non-crossing Partitions in Surprising Locations</a>, American Mathematical Monthly 113 (2006) 598-610.
%H A134264 M. Mendez, <a href="https://arxiv.org/abs/1610.03602">Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees</a>, arXiv:1610.03602 [math.CO], 2016, pp. 33-34 Example 10.
%H A134264 J. Pitman and R. Stanley, <a href="https://arxiv.org/abs/math/9908029">A polytope related to empirical distributions, plane trees, parking functions, and the associahedron</a>, arXiv:math/9908029 [math.CO], 1999.
%H A134264 A. Rattan, <a href="http://econ109.econ.bbk.ac.uk/arattan/mypubs/thesis.pdf">Parking functions and related combinatorial structures</a>, Master's thesis, Univ. of Waterloo, 2014.
%H A134264 A. Schuetz and G. Whieldon, <a href="http://arxiv.org/abs/1401.7194">Polygonal Dissections and Reversions of Series</a>, arXiv:1401.7194 [math.CO], 2014.
%H A134264 R. Simion, <a href="https://doi.org/10.1016/S0012-365X(99)00273-3">Noncrossing partitions</a>, Discrete Mathematics, Vol. 217, Issues 1-3, pp. 367-409, 2000.
%H A134264 R. Speicher, <a href="https://arxiv.org/abs/1908.08125">Lecture Notes on "Free Probability Theory"</a>, arXiv:1908.08125 [math.OA], 2019.
%H A134264 R. Stanley, <a href="http://www-math.mit.edu/~rstan/papers/nc.pdf">Parking Functions and Noncrossing Partitions</a>, 1996.
%H A134264 Jin Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Wang/wang53.html">Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.
%H A134264 Wikipedia, <a href="http://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>
%H A134264 J. Zhou, <a href="https://arxiv.org/abs/2008.03407">On emergent geometry of the Gromov-Witten theory of quintic Calabi-Yau threefold</a>, arXiv:2008.03407 [math-ph], 2020.
%H A134264 <a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>
%F A134264 For j>1, there are P(j,m;a...) = j! / [ (j-m)! (a_1)! (a_2)! ... (a_(j-1))! ] permutations of h_0 through h_(j-1) in which h_0 is repeated (j-m) times; h_1, repeated a_1 times; and so on with a_1 + a_2 + ... + a_(j-1) = m.
%F A134264 If, in addition, a_1 + 2 * a_2 + ... + (j-1) * a_(j-1) = j-1, then each distinct combination of these arrangements is correlated with a partition of j-1.
%F A134264 T(j,k) is [ P(j,m;a...) / j ] for the k-th partition of j-1 as described in the comments.
%F A134264 For example from g(t) above, T(5,4) = (5! / ((5-3)! * 2!)) / 5 = 6 for the 4th partition under n=5-1=4 with m=3 parts in A&S.
%F A134264 From _Tom Copeland_, Sep 30 2011: (Start)
%F A134264 Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}
%F A134264 = [(h_0)-1+:1/(1-h.*x):]^2 / {(h_0)-:[h.x/(1-h.x)]^2:}
%F A134264 = [(h_0)+(h_1)x+(h_2)x^2+...]^2 / [(h_0)-(h_2)x^2-2(h_3)x^3-3(h_4)x^4-...], where :" ": denotes umbral evaluation of the expression within the colons and h. is an umbral coefficient.
%F A134264 Then for the partition polynomials of A134264,
%F A134264 Poly[n;h_0,...,h_(n-1)]=(1/n!)(W(x)*d/dx)^n x, evaluated at x=0, and the compositional inverse of f(t) is g(t) = exp(t*W(x)*d/dx) x, evaluated at x=0. Also, dg(t)/dt = W(g(t)), and g(t) gives A001263 with (h_0)=u and (h_n)=1 for n>0 and A000108 with u=1.
%F A134264 (End)
%F A134264 From _Tom Copeland_, Oct 20 2011: (Start)
%F A134264 With exp(x* PS(.,t)) = exp(t*g(x)) = exp(x*W(y)d/dy) exp(t*y) eval. at y=0, the raising (creation) and lowering (annihilation) operators defined by R PS(n,t) = PS(n+1,t) and L PS(n,t) = n*PS(n-1,t) are
%F A134264 R = t*W(d/dt) = t*((h_0) + (h_1)d/dt + (h_2)(d/dt)^2 + ...)^2 / ((h_0) - (h_2)(d/dt)^2 - 2(h_3)(d/dt)^3 - 3(h_4)(d/dt)^4 + ...), and
%F A134264 L = (d/dt)/h(d/dt) = (d/dt) 1/((h_0) + (h_1)*d/dt + (h_2)*(d/dt)^2 + ...)
%F A134264 Then P(n,t) = (t^n/n!) dPS(n,z)/dz eval. at z=0 are the row polynomials of A134264. (Cf. A139605, A145271, and link therein to Mathemagical Forests for relation to planted trees on p. 13.)
%F A134264 (End)
%F A134264 Using the formalism of A263634, the raising operator for the partition polynomials of this array with h_0 = 1 begins as R = h_1 + h_2 D + h_3 D^2/2! + (h_4 - h_2^2) D^3/3! + (h_5 - 5 h_2 h_3) D^4/4! + (h_6 + 5 h_2^3 - 7 h_3^2 - 9 h_2 h_4) D^5/5! + (h_7 - 14 h_2 h_5 + 56 h_2^2 h_3) D^6/6! + ... with D = d/d(h_1). - _Tom Copeland_, Sep 09 2016
%F A134264 Let h(x) = x/f^{-1}(x) = 1/[1-(c_2*x+c_3*x^2+...)], with c_n all greater than zero. Then h_n are all greater than zero and h_0 = 1. Determine P_n(t) from exp[t*f^{-1}(x)] = exp[x*P.(t)] with f^{-1}(x) = x/h(x) expressed in terms of the h_n (cf. A133314 and A263633). Then P_n(b.) = 0 gives a recursion relation for the inversion polynomials of this entry a_n = b_n/n! in terms of the lower order inversion polynomials and P_j(b.)P_k(b.) = P_j(t)P_k(t)|_{t^n = b_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)]. - _Tom Copeland_, Feb 09 2018
%F A134264 A raising operator for the partition polynomials with h_0 = 1 regarded as a Sheffer Appell sequence in h_1 is described in A249548. - _Tom Copeland_, Jul 03 2018
%e A134264 1) With f(t) = t / (t-1), then h(t) = -(1-t), giving h_0 = -1, h_1 = 1 and h_n = 0 for n>1. Then g(t) = -t - t^2 - t^3 - ... = t / (t-1).
%e A134264 2) With f(t) = t*(1-t), then h(t) = 1 / (1-t), giving h_n = 1 for all n. The compositional inverse of this f(t) is g(t) = t*A(t) where A(t) is the o.g.f. for the Catalan numbers; therefore the sum over k of T(j,k), i.e., the row sum, is the Catalan number A000108(j-1).
%e A134264 3) With f(t) = (e^(-a*t)-1) / (-a), h(t) = Sum_{n>=0} Bernoulli(n) * (-a*t)^n / n! and g(t) = log(1-a*t) / (-a) = Sum_{n>=1} a^(n-1) * t^n / n. Therefore with h_n = Bernoulli(n) * (-a)^n / n!, Sum_{permutations s with s(1)+s(2)+...+s(j)=j-1} h_s(1) * h_s(2) * ... * h_s(j) = j * Sum_{k=1..(# of partitions for j-1)} T(j,k) * H(j-1,k ; h_0,h_1,...) = a^(j-1). Note, in turn, Sum_{a=1..m} a^(j-1) = (Bernoulli(j,m+1) - Bernoulli(j)) / j for the Bernoulli polynomials and numbers, for j>1.
%e A134264 4) With f(t,x) = t / (x-1+1/(1-t)), then h(t,x) = x-1+1/(1-t), giving (h_0)=x and (h_n)=1 for n>1. Then g(t,x) = (1-(1-x)*t-sqrt(1-2*(1+x)*t+((x-1)*t)^2)) / 2, a shifted o.g.f. in t for the Narayana polynomials in x of A001263.
%e A134264 5) With h(t)= o.g.f. of A075834, but with A075834(1)=2 rather than 1, which is the o.g.f. for the number of connected positroids on [n] (cf. Ardila et al., p. 25), g(t) is the o.g.f. for A000522, which is the o.g.f. for the number of positroids on [n]. (Added Oct 13 2014 by author.)
%e A134264 6) With f(t,x) = x / ((1-t*x)*(1-(1+t)*x)), an o.g.f. for A074909, the reverse face polynomials of the simplices, h(t,x) = (1-t*x) * (1-(1+t)*x) with h_0=1, h_1=-(1+2*t), and h_2=t*(1+t), giving as the inverse in x about 0 the o.g.f. (1+(1+2*t)*x-sqrt(1+(1+2*t)*2*x+x^2)) / (2*t*(1+t)*x) for signed A033282, the reverse face polynomials of the Stasheff polytopes, or associahedra. Cf. A248727. (Added Jan 21 2015 by author.)
%e A134264 7) With f(x,t) = x / ((1+x)*(1+t*x)), an o.g.f. for the polynomials (-1)^n * (1 + t + ... + t^n), h(t,x) = (1+x) * (1+t*x) with h_0=1, h_1=(1+t), and h_2=t, giving as the inverse in x about 0 the o.g.f. (1-(1+t)*x-sqrt(1-2*(1+t)*x+((t-1)*x)^2)) / (2*x*t) for the Narayana polynomials A001263. Cf. A046802. (Added Jan 24 2015 by author.)
%e A134264 From _Gus Wiseman_, Feb 15 2019: (Start)
%e A134264 Triangle begins:
%e A134264 1
%e A134264 1
%e A134264 1 1
%e A134264 1 3 1
%e A134264 1 4 2 6 1
%e A134264 1 5 5 10 10 10 1
%e A134264 1 6 6 3 15 30 5 20 30 15 1
%e A134264 1 7 7 7 21 42 21 21 35 105 35 35 70 21 1
%e A134264 Row 5 counts the following non-crossing set partitions:
%e A134264 {{1234}} {{1}{234}} {{12}{34}} {{1}{2}{34}} {{1}{2}{3}{4}}
%e A134264 {{123}{4}} {{14}{23}} {{1}{23}{4}}
%e A134264 {{124}{3}} {{12}{3}{4}}
%e A134264 {{134}{2}} {{1}{24}{3}}
%e A134264 {{13}{2}{4}}
%e A134264 {{14}{2}{3}}
%e A134264 (End)
%t A134264 Table[Binomial[Total[y],Length[y]-1]*(Length[y]-1)!/Product[Count[y,i]!,{i,Max@@y}],{n,7},{y,Sort[Sort/@IntegerPartitions[n]]}] (* _Gus Wiseman_, Feb 15 2019 *)
%o A134264 (PARI)
%o A134264 C(v)={my(n=vecsum(v), S=Set(v)); n!/((n-#v+1)!*prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!))}
%o A134264 row(n)=[C(Vec(p)) | p<-partitions(n-1)]
%o A134264 { for(n=1, 7, print(row(n))) } \\ _Andrew Howroyd_, Feb 01 2022
%Y A134264 (A001263,A119900) = (reduced array, associated g(x)). See A145271 for meaning and other examples of reduced and associated.
%Y A134264 Other orderings are A125181 and A306438.
%Y A134264 Cf. A119900 (e.g.f. for reduced W(x) with (h_0)=t and (h_n)=1 for n>0).
%Y A134264 Cf. A248927 and A248120, "scaled" versions of this Lagrange inversion.
%Y A134264 Cf. A091867 and A125181, for relations to lattice paths and trees.
%Y A134264 Cf. A000045, A000108, A000957, A001764, A000522, A005043, A007317, A033282,A036038, A046802, A074909, A075834, A104597, A145271, A248727.
%Y A134264 Cf. A249548 for use of Appell properties to generate the polynomials.
%Y A134264 Cf. A133314, A049019, A019538, A127671, and A008292 for relations to permutahedra, Eulerians.
%Y A134264 Cf. A263634, A263633, A000041, A000110, A001263, A016098, A124794, A133437.
%Y A134264 Cf. A006013.
%Y A134264 Cf. A000521, A091406.
%Y A134264 Cf. A350499
%K A134264 nonn,tabf
%O A134264 1,6
%A A134264 _Tom Copeland_, Jan 14 2008
%E A134264 Added explicit t^6, t^7, and t^8 polynomials and extended initial table to include the coefficients of t^8. - _Tom Copeland_, Sep 14 2016
%E A134264 Title modified by _Tom Copeland_, May 28 2018
%E A134264 More terms from _Gus Wiseman_, Feb 15 2019
%E A134264 Title modified by _Tom Copeland_, Sep 10 2019