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A305749
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T(n,k) is the number of achiral color patterns (set partitions) in a row or loop of length n with k or fewer colors (sets).
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10
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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, 18, 8, 1, 1, 2, 3, 7, 12, 27, 27, 16, 1, 1, 2, 3, 7, 12, 30, 43, 54, 16, 1, 1, 2, 3, 7, 12, 31, 55, 107, 81, 32, 1, 1, 2, 3, 7, 12, 31, 58, 141, 171, 162, 32, 1, 1, 2, 3, 7, 12, 31, 59, 159, 266, 427, 243, 64, 1, 1, 2, 3, 7, 12, 31, 59, 163, 312, 688, 683, 486, 64, 1
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OFFSET
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1,5
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COMMENTS
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An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABC are equivalent, as are AAABB and BBBAA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a loop are equivalent, so for loops AAABCB = BAAABC = CBAAAB.
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..k} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0 <= n <= 1 & n==k].
T(n,k) = Sum_{j=1..k} A304972(n,j).
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EXAMPLE
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The array begins at T(1,1):
1 1 1 1 1 1 1 1 1 1 1 1 1 ...
1 2 2 2 2 2 2 2 2 2 2 2 2 ...
1 2 3 3 3 3 3 3 3 3 3 3 3 ...
1 4 6 7 7 7 7 7 7 7 7 7 7 ...
1 4 9 11 12 12 12 12 12 12 12 12 12 ...
1 8 18 27 30 31 31 31 31 31 31 31 31 ...
1 8 27 43 55 58 59 59 59 59 59 59 59 ...
1 16 54 107 141 159 163 164 164 164 164 164 164 ...
1 16 81 171 266 312 334 338 339 339 339 339 339 ...
1 32 162 427 688 883 963 993 998 999 999 999 999 ...
1 32 243 683 1313 1774 2069 2169 2204 2209 2210 2210 2210 ...
1 64 486 1707 3407 5103 6119 6634 6789 6834 6840 6841 6841 ...
1 64 729 2731 6532 10368 13524 15080 15790 15975 16026 16032 16033 ...
a(n) are the terms of this array read by antidiagonals.
For T(4,3)=6, the achiral pattern rows are AAAA, AABB, ABAB, ABBA, ABBC, and ABCA. The achiral pattern loops are AAAA, AAAB, AABB, ABAB, AABC, and ABAC.
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MATHEMATICA
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Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] +
Ach[n-2, k-1] + Ach[n-2, k-2]]; (* A304972 *)
Table[Sum[Ach[n, j], {j, 1, k - n + 1}], {k, 1, 15}, {n, 1, k}] // Flatten
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CROSSREFS
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Columns converge to the right to A080107.
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KEYWORD
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AUTHOR
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STATUS
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approved
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