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A326390
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The number of ways of seating n people around a table for the second time so that k pairs are maintained. T(n,k) read by rows.
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4
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1, 0, 1, 0, 0, 2, 0, 0, 0, 6, 0, 0, 16, 0, 8, 10, 0, 50, 50, 0, 10, 36, 144, 180, 240, 108, 0, 12, 322, 980, 1568, 1274, 686, 196, 0, 14, 2832, 8704, 11840, 10240, 4832, 1536, 320, 0, 16, 27954, 81000, 108054, 85050, 43902, 13446, 2970, 486, 0, 18, 299260, 834800, 1071700, 828400, 416200, 141520, 31000, 5200, 700, 0, 20
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OFFSET
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0,6
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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.
Sum of each row is equal to n!.
Weighted average of each row using k as weights converges to 2 for large n and is given with following formula: (Sum_{k} T(n,k)*k)/n! = 2/(n-1) + 2 (conjectured).
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LINKS
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FORMULA
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T(n,n) = 2*n for n > 2;
T(n,n-1) = 0 for n > 1;
T(n,n-2) = n^2*(n-3) for n > 3 (conjectured);
T(n,n-3) = (3/4)*n^4 + 6*n^3 + (2/3)*n^2 - 14*n + 6 for n > 4 (conjectured);
T(n,n-4) = (25/12)*n^5 + (73/6)*n^4 + (5/4)*n^3 - (253/6)*n^2 + (152/3)*n - 24 for n > 5 (conjectured);
T(n,n-5) = (52/15)*n^6 + (77/3)*n^5 + 14*n^4 - (194/3)*n^3 + (4628/15)*n^2 - 273*n + 130 for n > 5 (conjectured);
T(n,n-6) = (707/120)*n^7 + (2093/40)*n^6 + (2009/40)*n^5 - (245/8)*n^4 + (78269/60)*n^3 - (18477/10)*n^2 + (21294/10)*n - 684 for n > 6 (conjectured).
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EXAMPLE
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Assuming initial order was {1,2,3,4,5} (therefore 1 and 5 forms 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) and reflection ({1,2,3,5,4} and {4,5,3,2,1} are also different) the total number of ways is 5*2*5 and therefore T(5,3)=50.
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 2
3 0 0 0 6
4 0 0 16 0 8
5 10 0 50 50 0 10
6 36 144 180 240 108 0 12
7 322 980 1568 1274 686 196 0 14
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PROG
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(Java) See Links section
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CROSSREFS
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Cf. A326397 (disregards reflection symmetry), A326404 (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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