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A326397
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Triangle T(n,k) read by rows: T(n,k) = number of ways of seating n people around a table for the second time so that k pairs are maintained. Reflected sequences are counted as one.
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4
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1, 0, 1, 0, 0, 1, 0, 0, 0, 3, 0, 0, 8, 0, 4, 5, 0, 25, 25, 0, 5, 18, 72, 90, 120, 54, 0, 6, 161, 490, 784, 637, 343, 98, 0, 7, 1416, 4352, 5920, 5120, 2416, 768, 160, 0, 8, 13977, 40500, 54027, 42525, 21951, 6723, 1485, 243, 0, 9, 149630, 417400, 535850, 414200, 208100, 70760, 15500, 2600, 350, 0, 10, 1737241, 4691654
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OFFSET
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0,10
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COMMENTS
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Definition requires "pairs" and for n=0 it is assumed that there is 1 way of seating 0 people around a table for the second time so that 0 pairs are maintained and 1 person forms only one pair with him/herself. Therefore T(0,0)=1, T(1,0)=0 and T(1,1)=1.
Weighted average of each row using k as weights converges to 2 for large n and is given by the following formula: (Sum_{k} T(n,k)*k)/(Sum_{k} T(n,k)) = 2/(n-1) + 2 (conjectured).
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LINKS
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FORMULA
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T(n,n) = n for n>2.
T(n,n-1) = 0 for n>1.
T(n,n-3) = 1/2*n^3 + 3/4*n^2 - 2 (conjectured);
T(n,n-3) = (2/3)*n^4 + 3*n^3 + (1/3)*n^2 - 7*n + 3 for n > 4 (conjectured);
T(n,n-4) = (25/24)*n^5 + (73/12)*n^4 + (5/8)*n^3 - (253/12)*n^2 + (76/3)*n - 12 for n > 5 (conjectured);
T(n,n-5) = (26/15)*n^6 + (77/6)*n^5 + 7*n^4 - (97/3)*n^3 + (2314/15)*n^2 - 273/2*n + 65 for n > 5 (conjectured);
T(n,n-6) = (707/240)*n^7 + (2093/80)*n^6 + (2009/80)*n^5 - (245/16)*n^4 + (78269/120)*n^3 - (18477/20)*n^2 + (10647/0)*n - 342 for n > 6 (conjectured).
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EXAMPLE
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Assuming the initial order was {1,2,3,4,5} (therefore 1 and 5 form a pair as first and last person are neighbors in case of round table) there are 5 sets of ways of seating them again so that 3 pairs are conserved: {1,2,3,5,4}, {2,3,4,1,5}, {3,4,5,2,1}, {4,5,1,3,2}, {5,1,2,4,3}. Since within each set we allow for rotation ({1,2,3,5,4} and {2,3,5,4,1} are different) but not reflection ({1,2,3,5,4} and {4,5,3,2,1} are counted as one sequence) the total number of ways is 5*5 and therefore T(5,3)=25.
Unfolded table with n individuals (rows) forming k pairs (columns):
0 1 2 3 4 5 6 7
0 1
1 0 1
2 0 0 1
3 0 0 0 3
4 0 0 8 0 4
5 5 0 25 25 0 5
6 18 72 90 120 54 0 6
7 161 490 784 637 343 98 0 7
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PROG
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(Java) See Links section.
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CROSSREFS
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Cf. A326390 (with reflection symmetry), A326404 (with reflection symmetry, but disregards circular symmetry), A326411 (disregards both circular and reflection symmetry).
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KEYWORD
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AUTHOR
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STATUS
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approved
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