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The list partition transform of a sequence a(n) for which a(0)=1 is illustrated by
b(0) = 1
b(1) = -a(1)
b(2) = -a(2) + 2 a(1)^2
b(3) = -a(3) + 6 a(2)a(1) - 6 a(1)^3
b(4) = -a(4) + 8 a(3)a(1) + 6 a(2)^2 - 36 a(2)*a(1)^2 + 24 a(1)^4
The unsigned coefficients are A049019 with a leading 1. The sign is dependent on the partition as evident from inspection (replace a(n)'s by -1).
Expressed umbrally,
exp(a(.)*x)*exp(b(.)*x)= 1 , i.e.,
(a(.)+b(.))^n = 1 for n=0, and 0 for all other values of n.
Expressed recursively,
b(0) = 1
b(n) = -sum(j=1,...,n) binomial(n,j)*a(j)*b(n-j)
which is conditionally self-inverse, i.e., the roles of a(k) and b(k) may be reversed with a(0) = b(0) = 1.
Expressed in matrix form, b(n) form the first column of B = matrix inverse of A .
A = Pascal matrix diagonally multipied by a(n) ,
i.e. A(n,k) = binomial(n,k)*a(n-k).
Some examples of reciprocal pairs of sequences under these operations are
1) A084358 and -A000262 with the first term set to 1.
2) (1,-1,0,0,...) and (0!,1!,2!,3!,...) with the unsigned associated matrices A128229 and A094587.
3) (1,-1,-1,-1,...) and A000670.
4) A000110 and A000587.
5) (1,-2,-2,0,0,0,...) and (0! c(1),1! c(2),2! c(3),3! c(4),...) where c(n) = A000129(n) with the associated matrices A110327 and A110330.
6) (1,-2,2,0,0,0,...) and (1!,2!,3!,4!,...).
7) Sequences of rising and signed lowering factorials form reciprocal pairs where a(n) = (-1)^n m!/(m-n)! and b(n) = (m-1+n)!/(m-1)! for m=0,1,2,... .
Denote the action of the list partition transform on the sequence a(.) or an invertible matrix M by LPT(a) = b or LPT(M)= M^(-1).
If the matrix equation M = exp(T) also holds, then exp[a(.)*T] * exp[b(.)*T] = exp[(a+b)*T] = Identity matrix because (a(.)+b(.))^n = delta(n) , the Kronecker delta.
Therefore [exp(a*T)]^(-1) = exp[b*T] = exp[LPT(a)*T] = LPT[exp(a*T)].
The fundamental Pascal (A007318), unsigned Lah (A105278), and associated Laguerre (A??????) matrices can be generated by exponentiation of special infinitesimal matrices (see A132440, A132710, and A132681) such that finding LPT(a) amounts to diagonally multiplying any of the fundamental matrices by a(.), followed by matrix inversion, and then extraction of the b(.) factors from the first column (simplest for the Pascal formulae above).
Conversely, the inverses of matrices formed by diagonally multiplying the three fundamental matrices by a(.) are given by diagonally multiplying the fundamental matrices by b(.) .
If LPT(M) is defined differently as application of the top formula to a(n) = M^n, then b(n) = (-M)^n , and the formalism could even be applied to more general sequences of matrices M(.), providing the reciprocal of exp[t*M(.)].
The group of fundamental lower triangular matrices M = exp(T) such that LPT[exp(a*T)] = exp[LPT(a)*T] = [exp[(a)*T]]^(-1) are obtained by infinitesimal generator matrices of the form T =
0;
t(0), 0;
0, t(1), 0;
0, 0, t(2), 0;
0, 0, 0. t(3), 0;
...
T^m has trivially vanishing terms except along the m'th subdiagonal, which is a sequence of generalized factorials:
[ t(0)*t(1)...t(m-2)*t(m-1) , t(1)*t(2)...t(m-1)*t(m) , t(2)*t(3)...t(m)*t(m+1) , ... ] .
Therefore the principal submatrices of T (given by setting t(j) = 0 for j>n-1) are nilpotent with at least [Tsub_n]^(n+1) = 0.
The general group of matrices GM[a(.)] = exp[a(.)*T] can also be obtained through diagonal multiplication of M = exp(T) by the sequence a(.), as in the Pascal matrix example above, and their inverses by diagonal multiplication by LPT(a) = b.
Weighted-mappings interpretation for the top partition equation:
Given n pre-nodes (Pre) and k post-nodes (Post), each Pre is connected to only one Post and each Post has at least one Pre connected to it (surjections or onto functions/maps). Weight each Post by -a(m) where m is the number of connections to the Post.
Weight each map by the product of the Post weights and multiply by the number of maps that share the same connectivity. Sum over the possible mappings for n Pre. The result is b(n).
E.g. b(3) = [ 3 Pre to 1 Post ] + [ 3 Pre to 2 Post ] + [ 3 Pre to 3 Post ]
= [1 map with 1 Post with 3 connections] + [ 6 maps with 1 Post with 2 connections and 1 Post with 1 connection] + [6 maps with 3 Post with 1 connection each]
= -a(3) + 6 * [-a(2)*-a(1)] + 6 * [-a(1)*-a(1)*-a(1)] .
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