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A326404 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. Rotated sequences are counted as one.

%I A326404 #18 Sep 08 2023 22:37:12
%S A326404 1,0,1,0,0,1,0,0,0,2,0,0,4,0,2,2,0,10,10,0,2,6,24,30,40,18,0,2,46,140,
%T A326404 224,182,98,28,0,2,354,1088,1480,1280,604,192,40,0,2,3106,9000,12006,
%U A326404 9450,4878,1494,330,54,0,2,29926,83480,107170,82840,41620,14152,3100,520,70,0,2
%N A326404 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. Rotated sequences are counted as one.
%C A326404 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 A326404 Sum of row n is equal to (n-1)! for n > 1.
%C A326404 Conjecture: The 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.
%H A326404 Witold Tatkiewicz, <a href="/A326404/b326404.txt">Rows n = 0..17 of triangle, flattened</a>
%H A326404 Witold Tatkiewicz, <a href="https://pastebin.com/GnzsQtnz">Link for Java program</a>.
%F A326404 T(n,n) = 2 for n > 2;
%F A326404 T(n,n-1) = 0 for n > 1.
%F A326404 Conjectures:
%F A326404 T(n,n-2) = n^2 + n - 2 for n > 3;
%F A326404 T(n,n-3) = (4/3)*n^3 + 2*n^2 - (16/3)*n + 2 for n > 4;
%F A326404 T(n,n-4) = (25/12)*n^4 + (23/6)*n^3 - (169/12)*n^2 + (85/6)*n - 6 for n > 5;
%F A326404 T(n,n-5) = (52/15)*n^5 + (25/3)*n^4 - (83/3)*n^3 + (221/3)*n^2 - (299/5)*n + 26 for n > 5;
%F A326404 T(n,n-6) = (707/120)*n^6 + (2037/120)*n^5 - (413/8)*n^4 + (2233/8)*n^3 - (5554/15)*n^2 + (3739/10)*n - 114 for n > 6.
%e A326404 Assuming the initial order was {1,2,3,4,5} (therefore 1 and 5 form a pair as the first and last persons are neighbors in the case of a 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 do not allow for circular symmetry (e.g., {1,2,3,5,4} and its rotation to form {2,3,5,4,1} are counted as one) but we allow reflection ({1,2,3,5,4} and {4,5,3,2,1} are considered distinct), the total number of ways is 5*2 and therefore T(5,3)=10.
%e A326404 Unfolded table with n individuals (rows) forming k pairs (columns):
%e A326404     0    1    2    3    4    5    6    7
%e A326404 0   1
%e A326404 1   0    1
%e A326404 2   0    0    1
%e A326404 3   0    0    0    2
%e A326404 4   0    0    4    0    2
%e A326404 5   2    0   10   10    0    2
%e A326404 6   6   24   30   40   18    0    2
%e A326404 7  46  140  224  182   98   28    0    2
%o A326404 (Java) See Links section
%Y A326404 Cf. A078603 (column k=0).
%Y A326404 Sum of n-th row is A000142(n-1) for n > 0.
%Y A326404 Cf. A326390 (accounting for rotation and reflection symmetry), A326397 (disregards reflection symmetry but allows rotation), A326411 (disregards both reflection and rotation symmetry).
%K A326404 nonn,tabl
%O A326404 0,10
%A A326404 _Witold Tatkiewicz_, Aug 01 2019