A326397 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.

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

Offset: 0

Author

Witold Tatkiewicz, Aug 01 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.

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).

Examples

			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
		

Links

• Witold Tatkiewicz, Rows n = 0..17 of triangle, flattened
• Witold Tatkiewicz, link for java program

Crossrefs

Cf. A001710 sum of each row.

Cf. A326390 (with reflection symmetry), A326404 (with reflection symmetry, but disregards circular symmetry), A326411 (disregards both circular and reflection symmetry).

Programs

Java

See Links section.

Formula

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).