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A056273 Word structures of length n using a 6-ary alphabet.

%I A056273 #92 Jan 05 2025 23:41:10
%S A056273 1,1,2,5,15,52,203,876,4111,20648,109299,601492,3403127,19628064,
%T A056273 114700315,676207628,4010090463,23874362200,142508723651,852124263684,
%U A056273 5101098232519,30560194493456,183176170057707,1098318779272060,6586964947803695,39510014478620232,237013033135668883
%N A056273 Word structures of length n using a 6-ary alphabet.
%C A056273 Set partitions of the n-set into at most 6 parts; also restricted growth strings (RGS) with six letters s(1),s(2),...,s(6) where the first occurrence of s(j) precedes the first occurrence of s(k) for all j < k. - _Joerg Arndt_, Jul 06 2011
%C A056273 Permuting the alphabet will not change a word structure. Thus aabc and bbca have the same structure.
%C A056273 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} Product_{j=1..i} (j(1+...+j)*) where Sum stands for union and Product for concatenation. - _Nelma Moreira_, Oct 10 2004
%C A056273 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
%C A056273 From _Gary W. Adamson_, Jul 06 2011: (Start)
%C A056273 Construct an infinite array of sequences representing word structures of length n using an N-ary alphabet as follows:
%C A056273 .
%C A056273   1, 1, 1,  1,  1,   1,   1,    1, ...; N=1, A000012
%C A056273   1, 2, 4,  8, 16,  32,  64,  128, ...; N=2, A000079
%C A056273   1, 2, 5, 14, 41, 122, 365, 1094, ...; N=3, A007051
%C A056273   1, 2, 5, 15, 51, 187, 715, 2795, ...; N=4, A007581
%C A056273   1, 2, 5, 15, 52, 202, 855, 3845, ...; N=5, A056272
%C A056273   1, 2, 5, 15, 52, 203, 876, 4111, ...; N=6, A056273
%C A056273   ...
%C 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; ...). (End)
%D A056273 M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
%H A056273 Muniru A Asiru, <a href="/A056273/b056273.txt">Table of n, a(n) for n = 0..1275</a>
%H A056273 Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H A056273 Olli-Samuli Lehmus, <a href="https://aaltodoc.aalto.fi/bitstream/handle/123456789/122822/master_Lehmus_Olli-Samuli_2023.pdf">Optimized Static Allocation of Signal Processing Tasks onto Signal Processing Cores</a>, Master's Thesis, Aalto Univ. (Finland, 2023). See p. 35.
%H A056273 Nelma Moreira and Rogerio Reis, <a href="http://www.dcc.fc.up.pt/Pubs/TR04/dcc-2004-07.ps.gz">dcc-2004-07.ps</a>
%H A056273 Nelma Moreira and Rogerio Reis, <a href="http://www.dcc.fc.up.pt/dcc/Pubs/TReports/TR04/dcc-2004-07.pdf">On the density of languages representing finite set partitions</a>, Technical Report DCC-2004-07, August 2004, DCC-FC& LIACC, Universidade do Porto.
%H A056273 Nelma Moreira and Rogerio Reis, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
%H A056273 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (16,-95,260,-324,144).
%F A056273 a(n) = Sum_{k=0..6} Stirling2(n, k).
%F A056273 For n > 0, a(n) = (1/6!)*(6^n + 15*4^n + 40*3^n + 135*2^n + 264). - _Vladeta Jovovic_, Aug 17 2003
%F A056273 From _Nelma Moreira_, Oct 10 2004: (Start)
%F A056273 For n > 0 and c = 6:
%F A056273 a(n) = (c^n)/c! + Sum_{k=0..c-2} ((k^n)/k!*(Sum_{j=2..c-k}(((-1)^j)/j!))).
%F A056273 a(n) = 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)! if c>1. For 2 <= k <= c: g(k, c) = g(k-1, c-1)/k if c>1.  (End)
%F A056273 G.f.: (1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1-x)*(1-2x)*(1-3x)*(1-4x)*(1-6x)). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 26 2009 [corrected by _R. J. Mathar_, Sep 16 2009] [Adapted to offset 0 by _Robert A. Russell_, Nov 06 2018]
%F A056273 G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=6. - _Robert A. Russell_, Apr 25 2018
%F A056273 E.g.f.: (265 + 264*exp(x) + 135*exp(x*2) + 40*exp(x*3) + 15*exp(x*4) + exp(6*x))/6!. - _Peter Luschny_, Nov 06 2018
%e A056273 For a(4) = 15, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABBC, ABCA, and ABCD; the 8 chiral patterns are the 4 pairs AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.
%p A056273 egf := (265+264*exp(x)+135*exp(x*2)+40*exp(x*3)+15*exp(x*4)+exp(6*x))/6!:
%p A056273 ser := series(egf,x,30): seq(n!*coeff(ser,x,n),n=0..22); # _Peter Luschny_, Nov 06 2018
%t A056273 Table[Sum[StirlingS2[n,k],{k,0,6}],{n,0,30}] (* or *) LinearRecurrence[ {16,-95,260,-324,144},{1,1,2,5,15,52},30] (* _Harvey P. Dale_, Jun 05 2015 *)
%o A056273 (PARI) Vec((1 - 15*x + 81*x^2 - 192*x^3 + 189*x^4 - 53*x^5)/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)*(1-6*x)) + O(x^30)) \\ _Michel Marcus_, Nov 07 2018
%o A056273 (GAP) List([0..25],n->Sum([0..6],k->Stirling2(n,k))); # _Muniru A Asiru_, Oct 30 2018
%o A056273 (Magma) [(&+[StirlingSecond(n, i): i in [0..6]]): n in [0..30]]; // _Vincenzo Librandi_, Nov 07 2018
%Y A056273 A row of the array in A278984 and A320955.
%Y A056273 Cf. A000351, A000400, A007581, A056272, A008290, A007051, A099263, A099262, A000110, A008277.
%Y A056273 Cf. A056325 (unoriented), A320936 (chiral), A305752 (chiral).
%K A056273 nonn,easy
%O A056273 0,3
%A A056273 _Marks R. Nester_
%E A056273 a(0)=1 prepended by _Robert A. Russell_, Nov 06 2018