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.

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, 42, 1, 7, 7, 7, 28, 56, 28, 28, 84, 252, 84, 210, 420, 462, 132, 1, 8, 8, 8, 4, 36, 72, 72, 36, 36, 120, 360, 180, 360, 30, 330, 1320, 660, 792, 1980, 1716, 429

Offset: 1

Author

Tom Copeland, Jan 01 2022

Keywords

• nonn
• tabf

Comments

Coefficients are listed in Abramowitz and Stegun order (A036036).

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.

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.

Examples

			Triangle begins:
  1;
  1, 1;
  1, 3, 2;
  1, 4, 2, 10,  5;
  1, 5, 5, 15, 15, 35, 14;
  ...
___________
The first few free cumulants in terms of the free moments are
  c_1 = m_1
  c_2 = m_2 - m_1^2
  c_3 = m_3 - 3 m_2 m_1 + 2 m_1^3
  c_4 = m_4 - 2 m_2^2 - 4m_3 m_1 + 10 m_2 m_1^2 - 5 m_1^4
  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
___________
Conversely, from A134264, these free moments in terms of the free cumulants are
  m_1 = c_1
  m_2 = c_2 + c_1^2
  m_3 = c_3 + 3 c_2 c_1 + c_1^3
  m_4 = c_4 + + 2 c_2^2 + 4 c_3 c_1 + 6 c_2 c_1^2 + c_1^4
  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
___________
		

Links

• Tom Copeland, Ruling the inverse universe, the inviscid Hopf-Burgers evolution equation: Compositional inversion, free probability, associahedra, diff ids, integrable hierarchies, and translation, 2022
• MathOverflow, Combinatorics for the action of Virasoro / Kac-Schwarz operators: partition polynomials of free probability theory, a MO question posed by Tom Copeland, 2021.
• J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Crossrefs

Cf. A000041, A000108, A001764, A002293, A006318, A036040, A060693, A086810, A088617, A111785, A127671, A133437, A133932, A134685, A134264, A276850.

Programs

PARI

mv(n)={eval(Str("'m",n))}
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}
Q(n)={polcoef(-x/serreverse(x*(1 + sum(k=1, n, -x^k*mv(k), O(x*x^n)))), n)}
row(n)={my(q=Q(n)); [Trm(q,Vec(v)) | v<-partitions(n)]}
{ for(n=1, 7, print(row(n))) } \\ Andrew Howroyd, Feb 01 2022

PARI

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

Formula

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

Extensions

Terms a(19) and beyond from Andrew Howroyd, Feb 01 2022