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.

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

Offset: 0

Author

Witold Tatkiewicz, Jul 03 2019

Keywords

• nonn
• tabl

Comments

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).

Examples

			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
		

Links

• Witold Tatkiewicz, Rows n = 0..17 of triangle, flattened
• Witold Tatkiewicz, Link for Java program

Crossrefs

Cf. A089222 (column k=0).

Cf. A000142 sum of each row.

Cf. A326397 (disregards reflection symmetry), A326404 (disregards circular symmetry), A326411 (disregards both circular and reflection symmetry).

Programs

Java

See Links section

Formula

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).