A361928 - OEIS

%I #12 Mar 31 2023 14:23:25

%S 1,2,2,2,3,3,3,4,4,4,3,5,5,5,5,3,6,6,6,6,6,3,6,7,7,7,7,7,4,7,8,8,8,8,

%T 8,8,4,7,9,9,9,9,9,9,9,4,8,10,10,10,10,10,10,10,10,4,8,11,11,11,11,11,

%U 11,11,11,11,4,8,12,12,12,12,12,12,12,12,12,12,4,9,12,13,13,13,13,13,13,13,13,13,13,4,9,13,14,14,14,14,14,14,14,14,14,14,14,4,9

%N Triangle read by rows: T(n,d) = number of non-adaptive group tests required to identify exactly d defectives among n items.

%C Arguably, the triangle should be flanked by zeros on both sides -- for all n, T(n,0)=0 and T(n,n)=0 -- but these are not included here.

%C T(n,d) is the smallest number of rows that an n-column matrix can have while remaining d-separable.

%C Observations:

%C T(n,n-1) = n-1.

%C T(n,d) <= T(n+1,d).

%C T(n,d) <= T(n,d+1) whenever d <= n-2.

%C T(n,d) <= T(n+1,d) <= T(n,d)+1 whenever d < n.

%C T(n,d) < T(n+1,d+1) whenever d < n.

%C T(n+2,k+1) = n+1 whenever T(n,k)=n-1.

%C T(n, ceil(n/2)) = n-1 for all n >= 1.

%C Elaqqad writes (see Links): "The only value of d for which T(n,d) is known completely is d=1, for which T(n,1)=ceil(lg n); the exact value even for d=2 is not known. Generally, the solutions to this problem are called 'testing designs', and the main considered ones are: (1) Set-packing designs or block designs; (2) Transversal designs; (3) Designs whose d-disjunct or d-separable matrices are directly constructed."

%H Elaqqad, <a href="https://math.stackexchange.com/questions/3195281/partition-a-set-into-g-groups-k-different-ways-such-that-no-pair-of-elements-i/3488985#3488985">Partition a set into g groups, k different ways, such that no pair of elements is ever in the same group together more than M times</a>, MathOverflow

%H Arthur O'Dwyer, <a href="https://github.com/Quuxplusone/wolves-and-sheep">Quuxplusone/wolves-and-sheep</a>, a program to find optimal testing designs by brute-force exhaustive search

%e Triangle begins:

%e   1;

%e   2, 2;

%e   2, 3,  3;

%e   3, 4,  4,  4;

%e   3, 5,  5,  5,  5;

%e   3, 6,  6,  6,  6,  6;

%e   3, 6,  7,  7,  7,  7,  7;

%e   4, 7,  8,  8,  8,  8,  8,  8;

%e   4, 7,  9,  9,  9,  9,  9,  9,  9;

%e   4, 8, 10, 10, 10, 10, 10, 10, 10, 10;

%e   ...

%e If we have 8 items, 3 of which are defective, we can identify the 3 defectives in 6 tests:

%e        Test 1.  T..TT...

%e        Test 2.  T....TT.

%e        Test 3.  .T.T.T..

%e        Test 4.  .T..T.T.

%e        Test 5.  ..T.TT..

%e        Test 6.  ..TT..T.

%e For example: If tests (1,2,3,4,5) are positive, then items (1,2,5) are the defectives. If tests (2,3,4,5,6) are positive, then items (6,7,8) are the defectives. If tests (2,4,5,6) are positive, then items (3,7,8) are the defectives.

%Y Cf A054961: A054961(i) is the smallest n such that T(n,2)=i.

%Y Cf A290492: A290492(i) is the smallest n such that T(n,3)=i.

%K nonn,tabl

%O 2,2

%A _Arthur O'Dwyer_, Mar 30 2023