cp's OEIS Frontend

A326390 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.

%I A326390 #52 Aug 06 2019 21:50:17
%S A326390 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,
%T A326390 12,322,980,1568,1274,686,196,0,14,2832,8704,11840,10240,4832,1536,
%U A326390 320,0,16,27954,81000,108054,85050,43902,13446,2970,486,0,18,299260,834800,1071700,828400,416200,141520,31000,5200,700,0,20
%N A326390 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.
%C A326390 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.
%C A326390 Sum of each row is equal to n!.
%C A326390 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).
%H A326390 Witold Tatkiewicz, <a href="/A326390/b326390.txt">Rows n = 0..17 of triangle, flattened</a>
%H A326390 Witold Tatkiewicz, <a href="https://pastebin.com/q8ZxekrZ">Link for Java program</a>
%F A326390 T(n,n) = 2*n for n > 2;
%F A326390 T(n,n-1) = 0 for n > 1;
%F A326390 T(n,n-2) = n^2*(n-3) for n > 3 (conjectured);
%F A326390 T(n,n-3) = (3/4)*n^4 + 6*n^3 + (2/3)*n^2 - 14*n + 6 for n > 4 (conjectured);
%F A326390 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);
%F A326390 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);
%F A326390 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).
%e A326390 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.
%e A326390 Unfolded table with n individuals (rows) forming k pairs (columns):
%e A326390     0    1    2    3    4    5    6    7
%e A326390 0   1
%e A326390 1   0    1
%e A326390 2   0    0    2
%e A326390 3   0    0    0    6
%e A326390 4   0    0   16    0    8
%e A326390 5  10    0   50   50    0   10
%e A326390 6  36  144  180  240  108    0   12
%e A326390 7 322  980 1568 1274  686  196    0   14
%o A326390 (Java) See Links section
%Y A326390 Cf. A089222 (column k=0).
%Y A326390 Cf. A000142 sum of each row.
%Y A326390 Cf. A326397 (disregards reflection symmetry), A326404 (disregards circular symmetry), A326411 (disregards both circular and reflection symmetry).
%K A326390 nonn,tabl
%O A326390 0,6
%A A326390 _Witold Tatkiewicz_, Jul 03 2019