Search: id:A133314
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%I A133314
%S A133314 1,1,1,2,1,6,6,1,8,6,36,24,1,10,20,60,90,240,120,1
%V A133314 1,-1,-1,2,-1,6,-6,-1,8,6,-36,24,-1,10,20,-60,-90,240,-120,-1
%N A133314 Coefficients of list partition transform.
%C A133314 The list partition transform of a sequence a(n) for which a(0)=1 is illustrated 
               by
%C A133314 b(0) = 1
%C A133314 b(1) = -a(1)
%C A133314 b(2) = -a(2) + 2 a(1)^2
%C A133314 b(3) = -a(3) + 6 a(2)a(1) - 6 a(1)^3
%C A133314 b(4) = -a(4) + 8 a(3)a(1) + 6 a(2)^2 - 36 a(2)*a(1)^2 + 24 a(1)^4
%C A133314 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).
%C A133314 Expressed umbrally,
%C A133314 exp(a(.)*x)*exp(b(.)*x)= 1, i.e.,
%C A133314 (a(.)+b(.))^n = 1 for n=0 and 0 for all other values of n.
%C A133314 Expressed recursively,
%C A133314 b(0) = 1
%C A133314 b(n) = -sum(j=1,...,n) binomial(n,j)*a(j)*b(n-j)
%C A133314 which is conditionally self-inverse, i.e., the roles of a(k) and b(k) 
               may be reversed with a(0) = b(0) = 1.
%C A133314 Expressed in matrix form, b(n) form the first column of B = matrix inverse 
               of A .
%C A133314 A = Pascal matrix diagonally multipied by a(n), i.e. A(n,k) = binomial(n,
               k)*a(n-k).
%C A133314 Some examples of reciprocal pairs of sequences under these operations 
               are
%C A133314 1) A084358 and -A000262 with the first term set to 1.
%C A133314 2) (1,-1,0,0,...) and (0!,1!,2!,3!,...) with the unsigned associated 
               matrices A128229 and A094587.
%C A133314 3) (1,-1,-1,-1,...) and A000670.
%C A133314 4) A000110 and A000587.
%C A133314 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.
%C A133314 6) (1,-2,2,0,0,0,...) and (1!,2!,3!,4!,...).
%C A133314 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,... .
%C A133314 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).
%C A133314 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.
%C A133314 Therefore [exp(a*T)]^(-1) = exp[b*T] = exp[LPT(a)*T] = LPT[exp(a*T)].
%C A133314 The fundamental Pascal (A007318), unsigned Lah (A105278) and associated 
               Laguerre 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).
%C A133314 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(.) .
%C A133314 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(.)].
%C A133314 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 =
%C A133314 0;
%C A133314 t(0), 0;
%C A133314 0, t(1), 0;
%C A133314 0, 0, t(2), 0;
%C A133314 0, 0, 0, t(3), 0;
%C A133314 ...
%C A133314 T^m has trivially vanishing terms except along the m'th subdiagonal, 
               which is a sequence of generalized factorials:
%C A133314 [ 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), 
               ... ] .
%C A133314 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.
%C A133314 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.
%C A133314 Weighted-mappings interpretation for the top partition equation:
%C A133314 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.
%C A133314 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).
%C A133314 E.g. b(3) = [ 3 Pre to 1 Post ] + [ 3 Pre to 2 Post ] + [ 3 Pre to 3 
               Post ]
%C A133314 = [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]
%C A133314 = -a(3) + 6 * [-a(2)*-a(1)] + 6 * [-a(1)*-a(1)*-a(1)] .
%Y A133314 Sequence in context: A095132 A028940 A048998 this_sequence A049019 A046651 
               A063007
%Y A133314 Adjacent sequences: A133311 A133312 A133313 this_sequence A133315 A133316 
               A133317
%K A133314 sign
%O A133314 0,4
%A A133314 Tom Copeland (tcjpn(AT)msn.com), Oct 18 2007, Oct 29 2007, Nov 16 2007

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