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A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets). 3

%I #13 Nov 04 2019 18:46:59

%S 1,1,1,1,2,1,1,2,2,1,1,2,3,4,1,1,2,3,6,4,1,1,2,3,7,9,8,1,1,2,3,7,11,

%T 26,10,1,1,2,3,7,12,39,53,20,1,1,2,3,7,12,42,103,146,30,1,1,2,3,7,12,

%U 43,123,367,369,56,1,1,2,3,7,12,43,126,503,1235,1002,94,1,1,2,3,7,12,43,127,539,2008,4439,2685,180,1,1,2,3,7,12,43,127,543,2304,8720,15935,7434,316,1,1,2,3,7,12,43,127,544,2356,11023,38365,58509,20441,596,1

%N Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets).

%C Two color patterns are equivalent if the colors are permuted. An oriented cycle counts each chiral pair as two.

%C Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.

%C In other words, the number of n-bead necklace structures using a maximum of k different colored beads. - _Andrew Howroyd_, Oct 30 2019

%D 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 Andrew Howroyd, <a href="/A320747/b320747.txt">Table of n, a(n) for n = 1..1275</a>

%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.

%F T(n,k) = (1/n)*Sum_{j=1..k} Sum_{d|n} phi(d)*A(d,n/d,j), where A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

%F T(n,k) = A320748(n,k) + A320742(n,k) = 2*A320748(n,k) - A305749(n,k) = A305749(n,k) + 2*A320742(n,k).

%e Array begins with T(1,1):

%e 1 1 1 1 1 1 1 1 1 1 1 1 ...

%e 1 2 2 2 2 2 2 2 2 2 2 2 ...

%e 1 2 3 3 3 3 3 3 3 3 3 3 ...

%e 1 4 6 7 7 7 7 7 7 7 7 7 ...

%e 1 4 9 11 12 12 12 12 12 12 12 12 ...

%e 1 8 26 39 42 43 43 43 43 43 43 43 ...

%e 1 10 53 103 123 126 127 127 127 127 127 127 ...

%e 1 20 146 367 503 539 543 544 544 544 544 544 ...

%e 1 30 369 1235 2008 2304 2356 2360 2361 2361 2361 2361 ...

%e 1 56 1002 4439 8720 11023 11619 11697 11702 11703 11703 11703 ...

%e 1 94 2685 15935 38365 54682 60499 61579 61684 61689 61690 61690 ...

%e 1 180 7434 58509 173609 284071 336447 349746 351619 351766 351772 351773 ...

%e For T(4,2)=4, the patterns are AAAA, AAAB, AABB, and ABAB.

%e For T(4,3)=6, the patterns are the above four, AABC and ABAC.

%t Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d, Adnk[d,n-1,k-#] &], Boole[n==0 && k==0]]

%t Table[Sum[DivisorSum[n, EulerPhi[#] Adnk[#,n/#,j] &], {j,k-n+1}]/n, {k,15}, {n,k}] // Flatten

%o (PARI) \\ R is A152175 as square matrix

%o R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}

%o T(n)={my(M=R(n)); for(i=2, n, M[,i] += M[,i-1]); M}

%o { my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ _Andrew Howroyd_, Nov 03 2019

%Y Partial row sums of A152175.

%Y Columns 1-6 are A057427, A000013, A002076, A056292, A056293, A056294.

%Y For increasing k, columns converge to A084423.

%Y Cf. A320748 (unoriented), A320742 (chiral), A305749 (achiral).

%K nonn,tabl,easy

%O 1,5

%A _Robert A. Russell_, Oct 21 2018

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