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A350499 Unsigned coefficients of free moment partition polynomials determining the free cumulants from the free moments of free probability theory. Irregular triangle with row lengths given by A000041, n >= 1.

%I A350499 #63 Apr 04 2025 22:43:55
%S A350499 1,1,1,1,3,2,1,4,2,10,5,1,5,5,15,15,35,14,1,6,6,3,21,42,7,56,84,126,
%T A350499 42,1,7,7,7,28,56,28,28,84,252,84,210,420,462,132,1,8,8,8,4,36,72,72,
%U A350499 36,36,120,360,180,360,30,330,1320,660,792,1980,1716,429
%N A350499 Unsigned coefficients of free moment partition polynomials determining the free cumulants from the free moments of free probability theory. Irregular triangle with row lengths given by A000041, n >= 1.
%C A350499 Coefficients are listed in Abramowitz and Stegun order (A036036).
%C A350499 Irregular triangular matrix of the unsigned coefficients of the free moment partition polynomials of free probability theory, for a single variable, that give the free formal cumulants given the free formal moments. This set of partition polynomials together with those of A134264 are the counterparts to the exp-log relations for the classical formal moments and cumulants associated with A036040 and A127671.
%C A350499 Associations with a compositional inverse pair of Laurent series, Kac-Schwarz operators of 2-D quantum theory, Virasoro / Witt / Heisenberg group actions, and KP and KdV integrable hierarchies are noted in references supplied in the MathOverflow link as well as a geometric combinatorial model based upon noncrossing partitions.
%H A350499 Tom Copeland, <a href="https://tcjpn.wordpress.com/2022/01/27/ruling-the-inverse-universe-the-inviscid-hopf-burgers-evolution-equation-compositional-inversion-associahedra-diff-ids-integrable-hierarchies-and-translation/">Ruling the inverse universe, the inviscid Hopf-Burgers evolution equation: Compositional inversion, free probability, associahedra, diff ids, integrable hierarchies, and translation</a>, 2022
%H A350499 MathOverflow, <a href="https://mathoverflow.net/questions/412573/combinatorics-for-the-action-of-virasoro-kac-schwarz-operators-partition-poly">Combinatorics for the action of Virasoro / Kac-Schwarz operators: partition polynomials of free probability theory</a>, a MO question posed by Tom Copeland, 2021.
%H A350499 J. Zhou, <a href="https://arxiv.org/abs/1405.5296">Quantum deformation theory of the Airy curve and the mirror symmetry of a point</a>, arXiv preprint arXiv:1405.5296 [math.AG], 2014.
%F A350499 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.
%F A350499 Row sums: big Schroeder numbers A006318.
%F A350499 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.
%F A350499 The coefficients of the highest order terms in m_1^n of the free moment partition polynomials are the signed Catalan numbers A000108.
%F A350499 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.,
%F A350499   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.
%F A350499 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.
%F A350499 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).
%F A350499 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.
%F A350499 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
%F A350499 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
%e A350499 Triangle begins:
%e A350499   1;
%e A350499   1, 1;
%e A350499   1, 3, 2;
%e A350499   1, 4, 2, 10,  5;
%e A350499   1, 5, 5, 15, 15, 35, 14;
%e A350499   ...
%e A350499 ___________
%e A350499 The first few free cumulants in terms of the free moments are
%e A350499   c_1 = m_1
%e A350499   c_2 = m_2 - m_1^2
%e A350499   c_3 = m_3 - 3 m_2 m_1 + 2 m_1^3
%e A350499   c_4 = m_4 - 2 m_2^2 - 4m_3 m_1 + 10 m_2 m_1^2 - 5 m_1^4
%e A350499   c_5 = m_5 - 5 m_2  m_3 - 5  m_4 m_1 + 15  m_2^2 m_1 + 15 m_3 m_1^2 - 35 m_2 m_1^3 + 14 m_1^5
%e A350499 ___________
%e A350499 Conversely, from A134264, these free moments in terms of the free cumulants are
%e A350499   m_1 = c_1
%e A350499   m_2 = c_2 + c_1^2
%e A350499   m_3 = c_3 + 3 c_2 c_1 + c_1^3
%e A350499   m_4 = c_4 + + 2 c_2^2 + 4 c_3 c_1 + 6 c_2 c_1^2 + c_1^4
%e A350499   m_5 = c_5 + 5 c_2 c_3 + 5 c_4 c_1 + 10 c_2^2 c_1 + 10 c_3 c_1^2  + 10 c_2 c_1^3 + c_1^5
%e A350499 ___________
%o A350499 (PARI)
%o A350499 mv(n)={eval(Str("'m",n))}
%o A350499 Trm(m,v)={my(S=Set(v)); for(i=1, #S, my(x=S[i]); m=polcoef(m, #select(y->y==x, v), mv(x))); m}
%o A350499 Q(n)={polcoef(-x/serreverse(x*(1 + sum(k=1, n, -x^k*mv(k), O(x*x^n)))), n)}
%o A350499 row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]}
%o A350499 { for(n=1, 7, print(row(n))) } \\ _Andrew Howroyd_, Feb 01 2022
%o A350499 (PARI)
%o A350499 C(v)={my(n=vecsum(v), S=Set(v)); (n+#v-2)!/(n-1)!/prod(i=1, #S, my(x=S[i]); (#select(y->y==x, v))!)}
%o A350499 row(n)=[C(Vec(p)) | p<-partitions(n)]
%o A350499 { for(n=1, 7, print(row(n))) } \\ _Andrew Howroyd_, Feb 01 2022
%Y A350499 Cf. A000041, A000108, A001764, A002293, A006318, A036040, A060693, A086810, A088617, A111785, A127671, A133437, A133932, A134685, A134264, A276850.
%K A350499 nonn,tabf
%O A350499 1,5
%A A350499 _Tom Copeland_, Jan 01 2022
%E A350499 Terms a(19) and beyond from _Andrew Howroyd_, Feb 01 2022