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A060223 Number of orbits of length n under the map whose periodic points are counted by A000670. 78

%I

%S 1,1,1,4,18,108,778,6756,68220,787472,10224702,147512052,2340963570,

%T 40527565260,760095923082,15352212731820,332228417589720,

%U 7668868648772700,188085259069430744,4884294069438337428,133884389812214097774,3863086904690670182596

%N Number of orbits of length n under the map whose periodic points are counted by A000670.

%C From _Gus Wiseman_, Oct 14 2016: (Start)

%C A finite sequence is normal if it spans an initial interval of positive integers. The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, (2 2 1) * (2 1 3) = (2 1 2 2 1 3). If Q is the set of compositions (finite sequences of positive integers) then (Q,*) is an Abelian group freely generated by a set P of prime sequences. The number of normal prime sequences of length n is equal to a(n). See example 2 and Mathematica program 2.

%C If N is the species (endofunctor over the category of finite sets and permutations) of unlabeled necklaces and N(S) represents the set of all non-isomorphic primitive necklaces of length n=|S|, then the numbers |N(S)| are equal to the numbers a(|S|) for any finite set S. This is because the number of orderless *-factorizations (see A034691 and A269134) of any finite sequence q is equal to the number of multiset partitions (see A007716 and A255906) of the multiset of prime factors of q. (End)

%H Alois P. Heinz, <a href="/A060223/b060223.txt">Table of n, a(n) for n = 0..200</a>

%H Yash Puri and Thomas Ward, <a href="http://www.fq.math.ca/Scanned/39-5/puri.pdf">A dynamical property unique to the Lucas sequence</a>, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.

%H Y. Puri and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%H T. Ward, <a href="http://web.archive.org/web/20081002082625/http://www.mth.uea.ac.uk/~h720/research/files/integersequences.html">Exactly realizable sequences</a>

%F a(n) = (1/n)* Sum_{d|n} mu(d)*A000670(n/d) for n > 0, where mu is A008683, the Moebius function. - Edited by _Michel Marcus_, Mar 30 2016

%F Let A = Sum_{q in P} Prod_i x_{q_i} = Sum_y c_y m(y) be the symmetric function whose coefficient of m(y) is equal to the number of permutations of the normal multiset [k]^y that belong to P, where the multiplicity of i in [k]^y is defined to be y_i. Then a(n) is the sum of c_y taken over all integer partitions of n. See example 3. - _Gus Wiseman_, Oct 14 2016

%F a(n) = Sum_{d|n} mu(d) * A019536(n/d) for n >= 1. - _Petros Hadjicostas_, Aug 19 2019

%e a(5) = 108 since A000670(5) is 541 and A000670(1) is 1, so there must be (541-1)/5 = 108 orbits of length 5.

%e From _Gus Wiseman_, Oct 14 2016: (Start)

%e The a(4) = 18 normal prime sequences are the columns:

%e [2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4]

%e [1 2 2 1 1 1 2 2 2 2 3 3 1 1 2 2 3 3]

%e [1 1 2 1 2 2 1 1 2 3 1 2 2 3 1 3 1 2]

%e [1 1 1 2 1 2 1 2 1 1 2 1 3 2 3 1 2 1].

%e The symmetric function A(x_1,x_2,x_3,...) expanded in terms of monomial symmetric functions m(y) (indexed by integer partitions y) is equal to:

%e A = m(1)+

%e m(11)+

%e 2m(21)+2m(111)+

%e m(22)+2m(31)+9m(211)+6m(1111)+

%e 4m(32)+2m(41)+18m(221)+12m(311)+48m(2111)+24m(11111)+

%e 3m(33)+4m(42)+2m(51)+14m(222)+60m(321)+15m(411)+180m(2211)+80m(3111)+300m(21111)+120m(111111)+... (End)

%t a[n_] := DivisorSum[n, MoebiusMu[#] HurwitzLerchPhi[1/2, -n/#, 0]/2 &] / n; a[0] = 1; Table[a[n], {n, 0, 30}] (* _Jean-Fran├žois Alcover_, Mar 30 2016 *)

%t thufbin[{},b_List]:=b;thufbin[a_List,{}]:=a;thufbin[a_List]:=a;

%t thufbin[{x_,a___},{y_,b___}]:=Switch[Ordering[If[x=!=y,{x,y},{thufbin[{a},{x,b}],thufbin[{x,a},{b}]}]],{1,2},Prepend[thufbin[{a},{y,b}],x],{2,1},Prepend[thufbin[{x,a},{b}],y]];

%t thufbin[a_List,b_List,c__List]:=thufbin[a,thufbin[b,c]];

%t priseqs[n_]:=Fold[Select,Tuples[Range[n],n],{Union[#]===Range[First[#]]&,Function[q,Select[Table[List[Take[q,{1,j}],Take[q,{j+1,n}]],{j,1,n-1}],thufbin@@Sort[#]===q&,1]==={}]}];

%t Table[Length[priseqs[n]],{n,1,7}] (* _Gus Wiseman_, Oct 14 2016 *)

%o (PARI) \\ here b(n) is A000670

%o b(n)={polcoeff(serlaplace(1/(2-exp(x+O(x*x^n)))), n)}

%o a(n)={if(n<1, n==0, sumdiv(n, d, moebius(d)*b(n/d))/n)} \\ _Andrew Howroyd_, Dec 12 2017

%Y Cf. A000670, A034691 (multisets of compositions), A269134, A007716, A277427, A215474, A255906.

%Y Row sums of A254040.

%K easy,nonn

%O 0,4

%A Thomas Ward (t.ward(AT)uea.ac.uk), Mar 21 2001

%E More terms from _Alois P. Heinz_, Jan 23 2015

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Last modified September 21 02:46 EDT 2020. Contains 337266 sequences. (Running on oeis4.)