A361928 Triangle read by rows: T(n,d) = number of non-adaptive group tests required to identify exactly d defectives among n items.

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, 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, 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

Offset: 2

Author

Arthur O'Dwyer, Mar 30 2023

Keywords

• nonn
• tabl

Comments

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.

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

Observations:

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

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

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

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

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

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

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

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."

Examples

			Triangle begins:
  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,  8,  8;
  4, 7,  9,  9,  9,  9,  9,  9,  9;
  4, 8, 10, 10, 10, 10, 10, 10, 10, 10;
  ...
If we have 8 items, 3 of which are defective, we can identify the 3 defectives in 6 tests:
       Test 1.  T..TT...
       Test 2.  T....TT.
       Test 3.  .T.T.T..
       Test 4.  .T..T.T.
       Test 5.  ..T.TT..
       Test 6.  ..TT..T.
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.
		

Links

• Elaqqad, 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, MathOverflow
• Arthur O'Dwyer, Quuxplusone/wolves-and-sheep, a program to find optimal testing designs by brute-force exhaustive search

Crossrefs

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

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