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COMMENTS
| Restricted growth strings (RGS) with letters s1,s2,..,s6 where the first occurrence of sj precedes the first ocurence of sk for all j<k; also set partitions of the n-set into at most 6 parts. [Joerg Arndt, Jul 06 2011]
Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.
Density of regular language L over {1,2,3,4,5,6}^* (i.e. number of strings of length n in L) described by regular expression with c=6: sum_{i=1..c}(prod_{j=1..i}(j(1+..+j)*) where sum stands for union and prod for concatenation. - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
Word structures of length n using an N-ary alphabet are generated by taking M^n* the vector [(N 1's),0,0,0,...], leftmost column term = a(n+1). In the case of A056273, the vector = [1,1,1,1,1,1,0,0,0,...]. As the vector approaches all 1's, the leftmost column terms approach A000110, the Bell sequence. - Gary W. Adamson, Jun 23 2011
Construct an infinite array of sequences representing word structures of length n using an N-ary alphabet as follows:
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1,.1,..1,...1,...1,...1,...1,....1..; N=1, A000012
1,.2,..4,...8,..16,..32,..64,..128..; N=2, A000079
1,.2,..5,..14,..41,.122,.365,.1094..; N=3, A007051
1,.2,..5,..15,..51,.187,.715,.2795..; N=4, A007581
1,.2,..5,..15,..52,.202,.855,.3845..; N=5, A056272
1,.2,..5,..15,..52,.203,.876,.4111..; N=6, A056273
...
The sequences tend to A000110. Finite differences of columns reinterpreted as rows generate A008277 as a triangle: (1; 1,1; 1,3,1; 1,7,6,1;...). - Gary W. Adamson, Jul 06 2011
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FORMULA
| a(n) = sum(k=1..6, stirling2(n, k) ).
(1/6!)*(6^n+15*4^n+40*3^n+135*2^n+264). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 17 2003
For c=6, a(n)= (c^n)/c!+sum_{k=1..c-2}((k^n)/k!*(sum_{j=2..c-k}(((-1)^j)/j!))) or = sum_{k=1..c}(g(k, c)*k^n) where g(1, 1)=1 g(1, c)=g(1, c-1)+((-1)^(c-1))/(c-1)!, c>1 g(k, c)=g(k-1, c-1)/k, for c>1 and 2<= k<= c - Nelma Moreira (nam(AT)ncc.up.pt), Oct 10 2004
G.f.: x*(1-14*x+68*x^2-135*x^3+91*x^4)/ ((1-x) * (6*x-1) * (4*x-1) * (3*x-1) * (2*x-1)) [From Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009]
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