

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.


4



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

0,6


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/(n1) + 2 (conjectured).


LINKS

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


FORMULA

T(n,n) = 2*n for n > 2;
T(n,n1) = 0 for n > 1;
T(n,n2) = n^2*(n3) for n > 3 (conjectured);
T(n,n3) = (3/4)*n^4 + 6*n^3 + (2/3)*n^2  14*n + 6 for n > 4 (conjectured);
T(n,n4) = (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,n5) = (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,n6) = (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).


EXAMPLE

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


PROG

(Java) See Links section


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).
Sequence in context: A132792 A136572 A262679 * A053203 A158360 A309746
Adjacent sequences: A326387 A326388 A326389 * A326391 A326392 A326393


KEYWORD

nonn,tabl


AUTHOR

Witold Tatkiewicz, Jul 03 2019


STATUS

approved



