A036036 -id:A036036 - cp's OEIS frontend

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.

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, 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, 168, 84, 168, 14, 70, 280, 140, 56, 140, 28, 1, 1, 9, 9, 9, 9, 36, 72

Offset: 1

Tom Copeland, Jan 14 2008

Coefficients are listed in Abramowitz and Stegun order (A036036).
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).
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.
Denoting h_n by (n') for brevity, to 8th order in t,
g(t) = t * (0')
+ t^2 * [ (0') (1') ]
+ t^3 * [ (0')^2 (2') + (0') (1')^2 ]
+ t^4 * [ (0')^3 (3') + 3 (0')^2 (1') (2') + (0') (1')^3 ]
+ t^5 * [ (0')^4 (4') + 4 (0')^3 (1') (3') + 2 (0')^3 (2')^2 + 6 (0')^2 (1')^2 (2') + (0') (1')^4 ]
+ 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 ]
+ 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]
+ 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 ]
+ ..., 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.
A125181 is an extended, reordered version of the above sequence, omitting the leading 1, with alternate interpretations.
If the coefficients of partitions with the same exponent for h_0 are summed within rows, A001263 is obtained, omitting the leading 1.
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
From Tom Copeland, Oct 28-29 2014: (Start)
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).
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,..).
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.
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.
(End)
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
From Tom Copeland, Nov 10 2014: (Start)
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.
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]
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.
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.
(End)
From Tom Copeland, Nov 12 2014: (Start)
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.)
(End)
From Tom Copeland, Nov 24 2014: (Start)
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).
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.
Abramowitz and Stegun give combinatorial interpretations of A036038 and relations to other number arrays.
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.
(End)
These partition polynomials are dubbed the Voiculescu polynomials on page 11 of the He and Jejjala link. - Tom Copeland, Jan 16 2015
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
(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
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
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
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
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
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
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
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

			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).
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).
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.
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.
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.)
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.)
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.)
From _Gus Wiseman_, Feb 15 2019: (Start)
Triangle begins:
   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
   1   7   7   7  21  42  21  21  35 105  35  35  70  21   1
Row 5 counts the following non-crossing set partitions:
  {{1234}}  {{1}{234}}  {{12}{34}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
            {{123}{4}}  {{14}{23}}  {{1}{23}{4}}
            {{124}{3}}              {{12}{3}{4}}
            {{134}{2}}              {{1}{24}{3}}
                                    {{13}{2}{4}}
                                    {{14}{2}{3}}
(End)

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).

Andrew Howroyd, Table of n, a(n) for n = 1..2087 (rows 1..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
A. Alexandrov, Enumerative geometry, tau-functions, and Heisenberg-Virasoro algebra, arXiv:hep-th/1404.3402v3, 2015 (p.22).
M. Ashelevich and W. Mlotkowski, Semigroups of distributions with linear Jacobi parameters, arxiv.org/abs/1001.1540v3 [math.CO], p. 9, 2011.
F. Ardila, F. Rincon, and L. Williams, Positroids and non-crossing partitions, arXiv preprint arXiv:1308.2698v2 [math.CO], 2013 (p. 25).
T. Banica, S. Belinschi, M. Capitaine, and B. Collins, Free Bessel laws, arXiv preprint arXiv:0710.5931 [math.PR], 2008.
Freddy Cachazo and Bruno Giménez Umbert, Connecting Scalar Amplitudes using The Positive Tropical Grassmannian, arXiv:2205.02722 [hep-th], 2022.
David Callan, Sets, Lists and Noncrossing Partitions , arXiv:0711.4841 [math.CO], 2008.
B. Collins, I. Nechita, and K. Zyczkowski, Random graph states, maximal flow and Fuss-Catalan distributions, arXiv preprint arXiv:1003.3075 [quant-ph], 2010.
Tom Copeland, Compositional inversion and Appell sequences, Nov 2, 2014.
Tom Copeland, The Hirzebruch criterion for the Todd class Dec. 14, 2014.
Tom Copeland, Appell polynomials, cumulants, noncrossing partitions, Dyck lattice paths, and inversion Dec. 23, 2014.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
Tom Copeland, In the Realm of Shadows: Umbral inverses and associahedra, noncrossing partitions, symmetric polynomials, and similarity transforms, 2019.
Tom Copeland, A Taste of Moonshine in Free Moments, 2022.
R. Dickau, Non-crossing partitions, Robert Dickau's website, 2012.
K. Ebrahimi-Fard, L. Foissy, J. Kock, and F. Patras, Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations, arXiv:1907.01190 [math.CO], 2019, p. 25.
K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, arXiv:1409.5664v2 [math.CO], 2015, p. 12.
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Y. He and V. Jejjala, Modular Matrix Models, arXiv:hep-th/0307293, 2003.
J. Jong, A. Hock, and R. Wulkenhaar Catalan tables and a recursion relation in noncommutative quantum field theory, arXiv preprint arXiv:1904.11231 [math-ph], 2019.
M. Josuat-Verges, F. Menous, J. Novelli, and J. Thibon, Noncommutative free cumulants, arxiv.org/abs/1604.04759 [math.CO], 2016.
Germain Kreweras, Sur les partitions non croisées d'un cycle, Discrete Math. 1 333-350 (1972).
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MathOverflow, Guises of the Noncrossing Partitions (NCPs), an MO question posed by Tom Copeland, 2019.
MathOverflow, Combinatorial/diagrammatic models for free moment partition polynomials, an MO question posed by Tom Copeland, 2021.
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M. Mendez, Combinatorial differential operators in: Faà di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees, arXiv:1610.03602 [math.CO], 2016, pp. 33-34 Example 10.
J. Pitman and R. Stanley, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, arXiv:math/9908029 [math.CO], 1999.
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Index entries for sequences related to Łukasiewicz

(A001263,A119900) = (reduced array, associated g(x)). See A145271 for meaning and other examples of reduced and associated.
Other orderings are A125181 and A306438.
Cf. A119900 (e.g.f. for reduced W(x) with (h_0)=t and (h_n)=1 for n>0).
Cf. A248927 and A248120, "scaled" versions of this Lagrange inversion.
Cf. A091867 and A125181, for relations to lattice paths and trees.
Cf. A000045, A000108, A000957, A001764, A000522, A005043, A007317, A033282,A036038, A046802, A074909, A075834, A104597, A145271, A248727.
Cf. A249548 for use of Appell properties to generate the polynomials.
Cf. A133314, A049019, A019538, A127671, and A008292 for relations to permutahedra, Eulerians.
Cf. A263634, A263633, A000041, A000110, A001263, A016098, A124794, A133437.
Cf. A006013.
Cf. A000521, A091406.
Cf. A350499

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 *)

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))!))}
row(n)=[C(Vec(p)) | p<-partitions(n-1)]
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

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.
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.
T(j,k) is [ P(j,m;a...) / j ] for the k-th partition of j-1 as described in the comments.
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.
From Tom Copeland, Sep 30 2011: (Start)
Let W(x) = 1/(df(x)/dx)= 1/{d[x/h(x)]/dx}
  = [(h_0)-1+:1/(1-h.*x):]^2 / {(h_0)-:[h.x/(1-h.x)]^2:}
  = [(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.
Then for the partition polynomials of 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.
(End)
From Tom Copeland, Oct 20 2011: (Start)
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
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
L = (d/dt)/h(d/dt) = (d/dt) 1/((h_0) + (h_1)*d/dt + (h_2)*(d/dt)^2 + ...)
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.)
(End)
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
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
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

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
Title modified by Tom Copeland, May 28 2018
More terms from Gus Wiseman, Feb 15 2019
Title modified by Tom Copeland, Sep 10 2019

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 15, 14, 18, 20, 24, 32, 13, 25, 21, 22, 27, 30, 28, 36, 40, 48, 64, 17, 35, 33, 26, 45, 50, 42, 44, 54, 60, 56, 72, 80, 96, 128, 19, 49, 55, 39, 34, 75, 63, 70, 66, 52, 81, 90, 100, 84, 88, 108, 120, 112, 144, 160, 192, 256

Offset: 0

Gus Wiseman, Apr 30 2020

First differs from A334435 at a(75) = 99, A334435(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(22)(31)(211)(1111)
  (5)(32)(41)(221)(311)(2111)(11111)

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Compositions under the same order are A124734 (triangle).
The version for reversed (weakly increasing) partitions is A185974.
The constructive version is A334301.
Ignoring length gives A334434, or A334437 for reversed partitions.
The dual version (sum/length/revlex) is A334438.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).
Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.

Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n]],{n,0,8}]

A001222(a(n)) = A036043(n).

A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 30, 16, 24, 36, 60, 210, 32, 48, 72, 120, 180, 420, 2310, 64, 96, 144, 240, 216, 360, 840, 900, 1260, 4620, 30030, 128, 192, 288, 480, 432, 720, 1680, 1080, 1800, 2520, 9240, 6300, 13860, 60060, 510510, 256, 384, 576, 960, 864, 1440, 3360

Offset: 0

Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jul 02 2001

Partitions are ordered first by sum. Then all partitions of n are viewed as exponent tuples on n variables and their corresponding monomials are ordered reverse lexicographically. This gives a canonical ordering: [] [1] [2,0] [1,1] [3,0,0] [2,1,0] [1,1,1] [4,0,0,0] [3,1,0,0] [2,2,0,0] [2,1,1,0] [1,1,1,1]... Rearrangement of A025487, A036035 etc.

Or, least integer of each prime signature; resorted in accordance with the integer partitions described in A080577. - Alford Arnold, Feb 13 2008

			Partition [2,1,1,1] for n=5 gives 2^2*3*5*7 = 420.
The sequence begins:
   1;
   2;
   4,  6;
   8, 12,  30;
  16, 24,  36,  60, 210;
  32, 48,  72, 120, 180, 420, 2310;
  64, 96, 144, 240, 216, 360,  840, 900, 1260, 4620, 30030;
  ...

Alois P. Heinz, Rows n = 0..30, flattened
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph Invariants Based on the Divides Relation and Ordered by Prime Signatures, arXiv:1405.5283 [math.NT], 2014.

Cf. A001222 (bigomega), A025487, A059901.
See A080576 Maple (graded reflected lexicographic) ordering.
See A080577 Mathematica (graded reverse lexicographic) ordering.
See A036036 "Abramowitz and Stegun" (graded reflected colexicographic) ordering.
See A036037 for graded colexicographic ordering.

with(combinat): A063008_row := proc(n) local e,w,r;
r := proc(L) local B,i; B := NULL;
for i from nops(L) by -1 to 1 do
B := B,L[i] od; [%] end:
w := proc(e) local i, m, p, P; m := infinity;
P := permute([seq(ithprime(i),i=1..nops(e))]);
for p in P do m := min(m,mul(p[i]^e[i],i=1..nops(e))) od end:
[seq(w(e), e = r(partition(n)))] end:
seq(print(A063008_row(i)),i=0..6); # Peter Luschny, Jan 23 2011
# second Maple program:
b:= (n, i)-> `if`(n=0 or i=1, [[1$n]], [map(x->
    [i, x[]], b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> map(x-> mul(ithprime(i)^x[i], i=1..nops(x)), b(n$2))[]:
seq(T(n), n=0..9);  # Alois P. Heinz, Sep 03 2019

row[n_] := Product[ Prime[k]^#[[k]], {k, 1, Length[#]}]& /@ IntegerPartitions[n]; Table[row[n], {n, 0, 8}] // Flatten (* Jean-François Alcover, Dec 10 2012 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {Table[1, {n}]},Join[ Prepend[#, i]& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := Product[Prime[i]^#[[i]], {i, 1, Length[#]}]& /@ b[n, n];
T /@ Range[0, 9] // Flatten (* Jean-François Alcover, Jun 09 2021, after Alois P. Heinz *)

bigomega(T(n,k)) = n. - Andrew Howroyd, Mar 28 2020

Partially edited by N. J. A. Sloane, May 15, at the suggestion of R. J. Mathar

Corrected and (minor) edited by Daniel Forgues, Jan 03 2011

A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 28, 25, 30, 40, 27, 36, 48, 64, 17, 26, 33, 44, 35, 42, 56, 50, 45, 60, 80, 54, 72, 96, 128, 19, 34, 39, 52, 55, 66, 88, 49, 70, 63, 84, 112, 75, 100, 90, 120, 160, 81, 108, 144, 192, 256

Offset: 0

Alford Arnold, Mar 31 2007

The tree begins (at height n, n >= 0, nodes represent partitions of n)
0:  1
1:  2
2:  3  4
3:  5  6  8
4:  7 10  9 12 16
5: 11 14 15 20 18 24 32
...
and hence differs from A114622.
Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - Daniel Forgues, Aug 07 2018
These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - Gus Wiseman, May 19 2020

			The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:
(1) pick an odd number (e.g., 135)
(2) calculate its prime factorization (135 = 5*3*3*3)
(3) note the least prime factor (LPF(135) = 3)
(4) note the index of the LPF (index(3) = 2)
(5) subtract one from the index (2-1 = 1)
(6) calculate the prime associated with the value in step five (prime(1) = 2)
(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
From _Daniel Forgues_, Aug 07 2018: (Start)
Partitions of 4 in graded reverse lexicographic order:
{4}: p_4 = 7;
{3,1}: p_3 * p_1 = 5 * 2 = 10;
{2,2}: p_2 * p_2 = 3^2 = 9;
{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
From _Gus Wiseman_, May 19 2020: (Start)
The sequence together with the corresponding partitions begins:
    1: ()            24: (2,1,1,1)         35: (4,3)
    2: (1)           32: (1,1,1,1,1)       42: (4,2,1)
    3: (2)           13: (6)               56: (4,1,1,1)
    4: (1,1)         22: (5,1)             50: (3,3,1)
    5: (3)           21: (4,2)             45: (3,2,2)
    6: (2,1)         28: (4,1,1)           60: (3,2,1,1)
    8: (1,1,1)       25: (3,3)             80: (3,1,1,1,1)
    7: (4)           30: (3,2,1)           54: (2,2,2,1)
   10: (3,1)         40: (3,1,1,1)         72: (2,2,1,1,1)
    9: (2,2)         27: (2,2,2)           96: (2,1,1,1,1,1)
   12: (2,1,1)       36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
   16: (1,1,1,1)     48: (2,1,1,1,1)       19: (8)
   11: (5)           64: (1,1,1,1,1,1)     34: (7,1)
   14: (4,1)         17: (7)               39: (6,2)
   15: (3,2)         26: (6,1)             52: (6,1,1)
   20: (3,1,1)       33: (5,2)             55: (5,3)
   18: (2,2,1)       44: (5,1,1)           66: (5,2,1)
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11731 (rows 0 <= n <= 26).
OEIS Wiki, Partitions#Orderings of partitions.
Wikiversity, Lexicographic and colexicographic order.

Cf. A080577 (the partitions), A252461, A114622, A128628, A215366 (sorted rows).
Row lengths are A000041.
Compositions under the same order are A066099.
The opposite version (sum/lex) is A334434.
The length-sensitive version (sum/length/revlex) is A334438.
The version for reversed (weakly increasing) partitions is A334436.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Sum of prime indices is A056239.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A036037, A185974, A211992, A334301, A334433, A334435, A334437, A334439.

b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
                b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := b[n, n];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

From Gus Wiseman, May 19 2020: (Start)
A001222(a(n)) = A238966(n).
A001221(a(n)) = A115623(n).
A056239(a(n)) = A036042(n).
A061395(a(n)) = A331581(n).
(End)

A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 5, 2, 3, 1, 4, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 3, 3, 2, 4, 1, 5, 2, 2, 2, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7, 3, 4, 2, 5, 1, 6, 2, 2, 3

Offset: 0

Gus Wiseman, Apr 30 2020

			The sequence of all reversed partitions begins:
  ()         (1,4)        (1,1,1,1,2)
  (1)        (1,2,2)      (1,1,1,1,1,1)
  (2)        (1,1,3)      (7)
  (1,1)      (1,1,1,2)    (3,4)
  (3)        (1,1,1,1,1)  (2,5)
  (1,2)      (6)          (1,6)
  (1,1,1)    (3,3)        (2,2,3)
  (4)        (2,4)        (1,3,3)
  (2,2)      (1,5)        (1,2,4)
  (1,3)      (2,2,2)      (1,1,5)
  (1,1,2)    (1,2,3)      (1,2,2,2)
  (1,1,1,1)  (1,1,4)      (1,1,2,3)
  (5)        (1,1,2,2)    (1,1,1,4)
  (2,3)      (1,1,1,3)    (1,1,1,2,2)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
                            0
                           (1)
                        (2) (1,1)
                    (3) (1,2) (1,1,1)
            (4) (2,2) (1,3) (1,1,2) (1,1,1,1)
  (5) (2,3) (1,4) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)
Showing partitions as their Heinz numbers (see A334435) gives:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A036043.
Lexicographically ordered reversed partitions are A026791.
The dual ordering (sum/length/lex) of reversed partitions is A036036.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Ignoring length gives A228531.
Sorting partitions by Heinz number gives A296150.
The version for compositions is A296774.
The dual ordering (sum/length/lex) of non-reversed partitions is A334301.
Taking Heinz numbers gives A334435.
The version for regular (non-reversed) partitions is A334439 (not A036037).
Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A028657 Triangle read by rows: T(n,k) = number of n-node graphs with k nodes in distinguished bipartite block, k = 0..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 7, 4, 1, 1, 5, 13, 13, 5, 1, 1, 6, 22, 36, 22, 6, 1, 1, 7, 34, 87, 87, 34, 7, 1, 1, 8, 50, 190, 317, 190, 50, 8, 1, 1, 9, 70, 386, 1053, 1053, 386, 70, 9, 1, 1, 10, 95, 734, 3250, 5624, 3250, 734, 95, 10, 1, 1, 11, 125, 1324, 9343, 28576, 28576, 9343, 1324, 125, 11, 1

Offset: 0

Vladeta Jovovic, Jun 16 2000

Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color; the color classes are not interchangeable.
Also the number of principal transversal matroids (also known as fundamental transversal matroids) of size n and rank k (originally enumerated by Brylawski). - Gordon F. Royle, Oct 30 2007
This sequence is also obtained if we read the array A(m,n) = number of inequivalent m X n binary matrices by antidiagonals, where equivalence means permutations of rows or columns (m>=0, n>=0) [Kerber]. - N. J. A. Sloane, Sep 01 2013

			The triangle T(n,k) begins:
  1;
  1,  1;
  1,  2,  1;
  1,  3,  3,  1;
  1,  4,  7,  4,  1;
  1,  5, 13, 13,  5,  1;
  1,  6, 22, 36, 22,  6,  1;
  ...
For example, there are 36 graphs on 6 nodes with a distinguished bipartite block with 3 nodes.
The array A(m,n) (m>=0, n>=0) (see Comments) begins:
  1 1  1    1     1      1        1         1           1 ...
  1 2  3    4     5      6        7         8           9 ...
  1 3  7   13    22     34       50        70          95 ...
  1 4 13   36    87    190      386       734        1324 ...
  1 5 22   87   317   1053     3250      9343       25207 ...
  1 6 34  190  1053   5624    28576    136758      613894 ...
  1 7 50  386  3250  28576   251610   2141733    17256831 ...
  1 8 70  734  9343 136758  2141733  33642660   508147108 ...
  1 9 95 1324 25207 613894 17256831 508147108 14685630688 ...
... - _N. J. A. Sloane_, Sep 01 2013

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Alois P. Heinz, Rows n = 0..45, flattened (first 20 rows from R. W. Robinson)
Thomas H. Brylawski, An affine representation for transversal geometries, Studies in Appl. Math. 54 (1975), no. 2, 143-160.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 1.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 1.
M. A. Harrison, On the number of classes of binary matrices, IEEE Trans. Computers, 22 (1973), 1048-1051.
Vladeta Jovovic, Binary matrices up to row and column permutations
A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
Carlos Simpson, Learning proofs for the classification of nilpotent semigroups, arXiv:2106.03015 [cs.LG], 2021.
Index to OEIS entries related to number of inequivalent matrices modulo permutation of row and columns.

Row sums give A049312.
A246106 is a very similar array.
Cf. A055080, A049312, A052265, A242093.
Diagonals of the array A(m,n) give A002724, A002725, A002728.
Rows (or columns) give A002623, A002727, A006148, A052264.
A(n,k) = A353585(2, n, k).

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
      coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
    end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
seq(seq(A(n, d-n), n=0..d), d=0..14); # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Union[ Flatten[ Table[ Function[ {p}, p + j*x^i] /@ b[n - i*j, i-1], {j, 0, n/i}]]]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[ Sum[GCD[i, j] * Coefficient[s, x, i] * Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[i^Coefficient[s, x, i] * Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i] * Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t)={sum(j=1, #q, gcd(t, q[j]))}
A(n, m)={my(s=0); forpart(q=m, s+=permcount(q)*polcoef(exp(sum(t=1, n, 2^K(q, t)/t*x^t) + O(x*x^n)), n)); s/m!}
{ for(r=0, 10, for(k=0, r, print1(A(r-k,k), ", ")); print) } \\ Andrew Howroyd, Mar 25 2020

\\ G(k,x) gives k-th column as rational function (see Jovovic link).
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
Fix(q,x)={my(v=divisors(lcm(Vec(q))), u=apply(t->2^sum(j=1, #q, gcd(t, q[j])), v)); 1/prod(i=1, #v, my(t=v[i]); (1-x^t)^(sum(j=1, i, my(d=t/v[j]); if(!frac(d), moebius(d)*u[j]))/t))}
G(m,x)={my(s=0); forpart(q=m, s+=permcount(q)*Fix(q,x)); s/m!}
T(n,k)={my(m=max(k, n-k)); polcoef(G(n-m, x + O(x*x^m)), m)} \\ Andrew Howroyd, Mar 26 2020

A028657(n,k)=A353585(2, n, k) \\ M. F. Hasler, May 01 2022

A(m,n) = Sum_{p in P(m), q in P(n)} 2^Sum_{i in p, j in q} gcd(i,j) / (N(p) N(q)) where P(m) are the partition of m (see e.g., A036036), N(p) = Product_{distinct parts x in p} x^m(x)*m(x)!, m(x) = multiplicity of x in p. [corrected by Anders Kaseorg, Oct 04 2024]

cp's OEIS Frontend

A334301 Irregular triangle read by rows where row k is the k-th integer partition, if partitions are sorted first by sum, then by length, and finally lexicographically.

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A036035 Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents.

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A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

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A063008 Canonical partition sequence (see A080577) encoded by prime factorization. The partition [p1,p2,p3,...] with p1 >= p2 >= p3 >= ... is encoded as 2^p1 * 3^p2 * 5^p3 * ... .

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A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

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A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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