A072605
Number of necklaces with n beads over an n-ary alphabet {a1,a2,...,an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w.
Original entry on oeis.org
1, 1, 2, 4, 13, 50, 270, 1641, 11945, 96784, 887982, 8939051, 99298354, 1195617443, 15619182139, 219049941201, 3293800835940, 52746930894774, 897802366250126, 16167544246362567, 307372573011579188, 6148811682561390279, 129164845357784003661
Offset: 0
- Alois P. Heinz, Table of n, a(n) for n = 0..200
- Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
- Eric Weisstein's world of Mathematics, Necklaces
-
neck[li:{__Integer}] := Module[{n, d}, n=Plus@@li; d=n-First[li]; Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times@@((li/#2)!)&, 0, Divisors[GCD@@li]]/n]; Table[ Plus@@(neck /@ IntegerPartitions[n]), {n, 24}]
-
a(n)={if(n==0, 1, my(p=prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n))); sumdiv(n, d, eulerphi(n/d)*d!*polcoeff(p,d))/n)} \\ Andrew Howroyd, Dec 20 2017
A261600
Number of primitive (aperiodic, or Lyndon) necklaces with n beads such that beads of a largest subset have label 0, beads of a largest remaining subset have label 1, and so on.
Original entry on oeis.org
1, 1, 1, 3, 11, 49, 265, 1640, 11932, 96780, 887931, 8939050, 99298073, 1195617442, 15619180497, 219049941148, 3293800823995, 52746930894773, 897802366153076, 16167544246362566, 307372573010691195, 6148811682561388635, 129164845357775064609
Offset: 0
a(3) = 3: 001, 012, 021.
a(4) = 11: 0001, 0011, 0012, 0021, 0102, 0123, 0132, 0213, 0231, 0312, 0321.
- Alois P. Heinz, Table of n, a(n) for n = 0..300
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
- Eric Weisstein's World of Mathematics, Necklace
- Wikipedia, Lyndon word
- Wikipedia, Necklace (combinatorics)
- Index entries for sequences related to necklaces
-
with(numtheory):
b:= proc(n, i, g, d, j) option remember; `if`(g>0 and gn, 0, binomial(n/j, i/j)*b(n-i, i, igcd(i, g), d, j)))))
end:
a:= n-> `if`(n=0, 1, add(add((f-> `if`(f=0, 0, f*b(n$2, 0, d, j)))(
mobius(j)), j=divisors(d)), d=divisors(n))/n):
seq(a(n), n=0..25);
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b[n_, i_, g_, d_, j_] := b[n, i, g, d, j] = If[g>0 && gn, 0, Binomial[n/j, i/j]*b[n-i, i, GCD[i, g], d, j]]]]]; a[n_] := If[n==0, 1, Sum[Sum[ Function[f, If[f==0, 0, f*b[n, n, 0, d, j]]][MoebiusMu[j]], {j, Divisors[ d]}], {d, Divisors[n]}]/n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)
A261599
Number of primitive (aperiodic, or Lyndon) necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.
Original entry on oeis.org
1, 1, 0, 1, 1, 3, 13, 24, 67, 252, 1795, 4038, 16812, 61750, 349806, 3485026, 10391070, 49433135, 240064988, 1282012986, 9167581934, 131550811985, 459677212302, 2707382738558, 14318807586215, 94084166753923, 601900541189696, 5894253303715121
Offset: 0
a(4) = 1: 0001.
a(5) = 3: 00001, 00011, 00101.
a(6) = 13: 000001, 000011, 000101, 000112, 000121, 000122, 001012, 001021, 001022, 001102, 001201, 001202, 010102.
a(7) = 24: 0000001, 0000011, 0000101, 0000111, 0000112, 0000121, 0000122, 0001001, 0001011, 0001012, 0001021, 0001022, 0001101, 0001102, 0001201, 0001202, 0010011, 0010012, 0010021, 0010022, 0010101, 0010102, 0010201, 0010202.
- Alois P. Heinz, Table of n, a(n) for n = 0..300
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
- F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
- Eric Weisstein's World of Mathematics, Necklace
- Wikipedia, Lyndon word
- Wikipedia, Necklace (combinatorics)
- Index entries for sequences related to necklaces
-
with(numtheory):
b:= proc(n, i, g, d, j) option remember; `if`(i*(i+1)/20
and gn, 0, binomial(n/j, i/j)*b(n-i, i-1, igcd(i, g), d, j))))
end:
a:= n-> `if`(n=0, 1, add(add((f-> `if`(f=0, 0, f*b(n$2, 0, d, j)))(
mobius(j)), j=divisors(d)), d=divisors(n))/n):
seq(a(n), n=0..30);
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a[0] = 1; a[n_] := With[{P = Product[1 + x^k/k!, {k, 1, n}] + O[x]^(n+1) // Normal}, DivisorSum[n, MoebiusMu[n/#]*#!*Coefficient[P, x, #]&]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, after Andrew Howroyd *)
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a(n)={if(n==0, 1, my(p=prod(k=1, n, (1+x^k/k!) + O(x*x^n))); sumdiv(n, d, moebius(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017
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