cp's OEIS Frontend

Denote this set of partition polynomials by [RT], the permutahedra polynomials of A133314 by [P], the refined Eulerian polynomials of A145271 by [E], and the Lagrange inversion polynomials of A134685 for e.g.f.s by [L]. Let the typically noncommutative product of two sets, e.g., [P][E], represent the substitution of the polynomials of [E] for the indeterminates of [P], i.e., a composition at the level of the indeterminates (see A356145 for examples). Let [I] be the substitutional identity transformation, and mark the substitutional inverse with the superscript -1. Then the following relations hold.

[RT] = [P][E] = [P][L][P] = [P]^{-1}[L][P] = [P][L][P]^{-1} since [P] is an involution, i.e., [P]^2 = [I], or [P] = [P]^{-1}, so [RT] and [L] are conjugate duals.

[RT]^{-1} = ([P][E])^{-1} = [E]^{-1}[P] = ([P][L][P])^{-1} = [P][L][P] = [RT], with [E]^{-1} = A356145, since [L] and [P] are involutions, so is [RT], i.e., [RT]^2 = [I].

RT_n(a_1,a_2,...,a_n) = D_{x=0}^n 1 / [ D_x f^{(-1)}(x)] for which D_x is the derivative w.r.t. x and the indeterminates are defined by 1 / [D_x f(x)] = 1 + a_1 x + a_2 x^2/2! + a_3 x^3/3! + ... with f(x) and f^{(-1)}(x) a compositional inverse pair of formal Taylor series, or e.g.f.s. This is the analytic equivalent of the algebraic relation [RT] = [P][E]. In words, the partition polynomials of row n (initial row is 0) is the n-th coefficient of the formal Taylor series of the reciprocal of the derivative of the compositional inverse of a function in terms of the Taylor series coefficients of the reciprocal of the derivative of that function. Note the correspondence with the analytic interpretation of [E]^{-1} of A356145, consistent with the algebraic identities above.

RT_n(a_1,a_2,...,a_n) = D_{x=0}^n f'(f^{(-1)}(x)) also, by the inverse function theorem, where the prime denotes differentiation with respect to the argument of the function.

With all a_k = (-1)^k, RT_0 = RT_1 = 1, otherwise RT_n = 0. This is determined with f(x) = e^{x}-1 and f^{(-1)}(x) = log(1+x).

With all a_k = 1, RT_0 = 1, RT_1 = -1, otherwise RT_n=0. This is determined with f(x) =1-e^{-x} and f^{(-1)}(x) = -log(1-x).

With all a_k = -1, RT_0 = 1 and RT_n = 2^(n-1) otherwise. This is determined with f(x) = (x - log(2-e^x))/2 and f^{(-1)}(x) = x - log(cosh(x)). (Careful, these are not the row sums of the absolute values of the numerical coefficients, which for the first ten polynomials are 1, 1, 2, 4, 8, 18, 40, 122, 446, and 2428.)

With a_k = k! 2^k, RT_0 = 1 and RT_n = -2*(2(n-1))! / (n-1)! = -2*n!*A000108(n-1) otherwise. This is determined with f(x) = x - x^2 and f^{(-1)}(x) = (1 - sqrt(1-4x))/2. Similar relations hold for the Fuss-Catalan sequences with f(x) = x - x^{m+1} for m > 1.

O.g.f.: C(x) = 1 + c_1 x + c_2 x^2 + ... = x / (x + m_1 x^2 + m_2 x^3 + m_3 x^4 + ...)^(-1) = x / M^(-1)(x), the shifted reciprocal of the compositional inverse of a power series M(x) = x + m_1 x^2 + m_2 x^3 + ..., the o.g.f. of the free moments m_n in free probability theory.

Row sums: big Schroeder numbers A006318.

Refinement of A060693 and A088617, i.e., by letting m_n = -t and removing all resulting signs, the elements of these two lower triangular matrices are generated.

The coefficients of the highest order terms in m_1^n of the free moment partition polynomials are the signed Catalan numbers A000108.

Taking the derivative with respect to the indeterminate m_1 generates the Lagrange inversion partition polynomials, with shifted indices, of A133437 and A111785 with an overall scale factor. These Lagrange inversion polynomials are the refined Euler characteristic polynomials of the associahedra. E.g.,

D_{m_1} c_5 = 5 (- m_4 + 3 m_2^2 + 6 m_3 m_1 - 21 m_2 m_1^2 + 14 m_1^4). An analogous differential formula that applies to the classical formal cumulants in relation to the permutahedra is stated in my 2012 comment in A127671.

The o.g.f. satisfies the partial differential equations D_{m_1} (x / C(x)) = -(1/3) D_x (x / C(x))^3 and D_{m_1} (C(x) / x) = D_x (x / C(x)), where D_{m_1} and D_x are the partial derivatives with respect to m_1 and x.

More generally, D_{m_n} (x / C(x)) = -(1/(n+2)) D_x (x / C(x))^{n+2), equivalent to D_{m_n} M^(-1)(x) = -(1/(n+2)) D_x (M^(-1)(x))^{n+2). Equations of this type are found in Zhou (see eqn. 44 on p. 11), characterizing the KdV hierarchy. These differential equations can be transformed into the inviscid Burgers-Hopf partial differential equation (see, e.g., A133437, A086810, A001764, A002293, A133932, A134685, and A276850).

The formal free cumulants when identified as the indeterminates of the noncrossing Lagrange inversion partition polynomials NCP_n(c_1,c_2,...,c_n) = m_n of A134264 (as in the example section) satisfy the partial differential equations D_{m_k} NCP_n(c_1, ..., c_n) = d(m_n)/dm_k = delta_{n-k}, where delta_{n} is the Kronecker delta which is zero for all integers n other than n = 0, for which it evaluates as unity. This provides a recursion method for determining the partial derivatives dc_n/dm_k from the partial derivatives dc_p/dm_k and cumulants c_p with k <= p < n. For example, dc_k/dm_j = 0 for j > k and dc_k/dm_k = 1, so dm_3/dm_2 = 0 = D_{m_2} (c_3 + 3 c_2 c_1 + c_1^3) = dc_3/dm_2 + 3 c_1 dc_2/dm_2 = dc_3/dm_2 + 3 c_1 , implying dc_3/dm_2 = -3 c_1 = -3 m_1.

T(n,k) = (n+j-2)!/((n-1)!*Product_{i>=1} s_i!), where (1*s_1 + 2*s_2 + ... = n) is the k-th partition of n and j = s_1 + s_2 + ... is the number of parts. - Andrew Howroyd, Feb 01 2022

Conjecture: free cumulants in terms of the free moments are R(n,1) for n > 0 where R(n,k) = R(n-1,k+1) - Sum_{j=1..n-1} R(j,k)*R(n-j,1) for n > 1, k > 0 with R(1,k) = m_k for k > 0. - Mikhail Kurkov, Mar 30 2025