

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.


4



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
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OFFSET

0,10


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/(n1) + 2 (conjectured).


LINKS

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


FORMULA

T(n,n) = n for n>2.
T(n,n1) = 0 for n>1.
T(n,n3) = 1/2*n^3 + 3/4*n^2  2 (conjectured);
T(n,n3) = (2/3)*n^4 + 3*n^3 + (1/3)*n^2  7*n + 3 for n > 4 (conjectured);
T(n,n4) = (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,n5) = (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,n6) = (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).


EXAMPLE

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


PROG

(Java) See Links section.


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).
Sequence in context: A243163 A209490 A346879 * A140577 A068606 A106153
Adjacent sequences: A326394 A326395 A326396 * A326398 A326399 A326400


KEYWORD

nonn,tabl


AUTHOR

Witold Tatkiewicz, Aug 01 2019


STATUS

approved



