A212360
Partition array a(n,k) with the total number of necklaces (C_n symmetry) with n beads, each available in n colors, with color signature given by the k-th partition of n in Abramowitz-Stegun(A-St) order.
Original entry on oeis.org
1, 2, 1, 3, 6, 2, 4, 12, 12, 36, 6, 5, 20, 40, 120, 180, 240, 24, 6, 30, 90, 60, 300, 1200, 320, 1200, 2700, 1800, 120, 7, 42, 126, 210, 630, 3150, 2100, 3150, 4200, 25200, 12600, 12600, 37800, 15120, 720, 8, 56, 224, 392, 280, 1176, 7056, 11760, 9072, 11760, 11760, 88200, 58800, 176400, 22260, 58800, 470400, 352800, 141120, 529200, 141120, 5040
Offset: 1
n\k 1 2 3 4 5 6 7 8 9 10 11
1 1
2 2 1
3 3 6 2
4 4 12 12 36 6
5 5 20 40 120 180 240 24
6 6 30 90 60 300 1200 320 1200 2700 1800 120
...
See the link for the rows n=1..15.
a(3,1)=3 because the 3 necklaces with 3 beads coming in 3 colors have the color multinomials (here monomials) c[1]^3=c[1]*c[1]*c[1], c[2]^3 and c[3]^3. The partition of 3 is 3, the color representative is c[1]^3, and the equivalence class with color signature from the partition 3 has the three given members.
a(3,2)=6 from the color signature 2,1 with the representative multinomial c[1]^2 c[2] with coefficient A212359(3,2)=1, the only 3-necklace cyclic(112) (taking j for the color c[j]), and A035206(3,2)=6 members of the whole color equivalence class: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332).
a(3,3)=2, color signature 1^3=1,1,1 with representative multinomial c[1]*c[2]*c[3] with coefficient A212359(3,3)=2 from the two necklaces cyclic(1,2,3) and cyclic (1,3,2). There are no other members in this class (A035206(3,3)=1).
The sum of row nr. 3 is 11=A056665(3). See the example given there with c[1]=R, c[2]=G and c[3]=B.
A213934
Triangle with entry a(n,m) giving the number of necklaces of n beads (C_N symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 3, 3, 6, 1, 3, 10, 12, 24, 1, 8, 31, 50, 60, 120, 1, 9, 71, 180, 300, 360, 720, 1, 22, 187, 815, 1260, 2100, 2520, 5040, 1, 29, 574, 2324, 6496, 10080, 16800, 20160, 40320, 1, 66, 1373, 9570, 32268, 58464, 90720, 151200, 181440, 362880
Offset: 1
n\m 1 2 3 4 5 6 7 8 9 10 ...
1 1
2 1 1
3 1 1 2
4 1 3 3 6
5 1 3 10 12 24
6 1 8 31 50 60 120
7 1 9 71 180 300 360 720
8 1 22 187 815 1260 2100 2520 5040
9 1 29 574 2324 6496 10080 16800 20160 40320
10 1 66 1373 9570 32268 58464 90720 151200 181440 362880
...
a(5,3) = 4 + 6 = 10, from A212359(5,4) + A212359(5,5), because k(5,3,1) = 4 and p(5,3) = 2.
a(2,1) = 1 because the partition [2] of n=2 with part number m=1 corresponds to the representative color multinomial (here monomial) c[1]^2=c[1]*c[1], and there is one such representative necklace. There is another necklace color monomial in this class of n=2 colors where only m=1 color is active: c[2]*c[2]. See the triangle entry A213935(2,1)=2.
a(3,1) = 1 from the color monomial representative c[1]^3. This class has 2 other members: c[2]^3 and c[3]^3. See A213935(3,1)=3.
In general a(n,1)=1 and A213935(n,1)=n from the partition [n] providing the color signature and a representative c[1]^n.
a(3,2)=1 from the representative color multinomial c[1]^2*c[2] (from the m=2 partition [2,1] of n=3) leading to just one representative necklace cyclic(112) (when one uses j for color c[j]). The whole class consists of A213935(3,2)=6 necklaces: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and cyclic(332).
a(3,3)=2. The representative color multinomial is c[1]*c[2]*c[3] (from the m=3 partition [1,1,1]). There are the two non-equivalent representative necklaces cyclic(1,2,3) and cyclic(1,3,2) which constitute already the whole class (A213935(3,3)=2).
a(4,2) = 3 from two representative color multinomials c[1]^3*c[2] and c[1]^2*c[2]^2 (from the two m=2 partitions of n=4: [3,1] and [2,2]). The first one has one representative necklace, namely cyclic(1112), the second one originates from two representative necklaces: cyclic(1122) and cyclic(1212). Together these are the 3 necklaces counted by a(4,2). The class with the first representative consists of 4*3=12 necklaces, when all 4 colors are used. The class of the second representative consists of 2*6=12 necklaces. Together they sum up to the 24 necklaces counted by A213935(4,2).
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b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
a226874[n_, k_] := If[n k == 0, If[n == k, 1, 0], n! b[n, 1, k]];
T[n_, k_] := (1/n) Sum[EulerPhi[n/d] a226874[d, k], {d, Divisors[n]}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 06 2018, after Alois P. Heinz and Andrew Howroyd *)
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\\ here U is A226874 as vector of polynomials.
U(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
C(n)={my(t=U(n)); vector(n, n, vector(n, k, (1/n)*sumdiv(n, d, eulerphi(n/d) * polcoeff(t[d+1], k))))}
{ my(t=C(10)); for(n=1, #t, print(t[n])) } \\ Andrew Howroyd, Dec 20 2017
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