OFFSET
1,4
COMMENTS
T(n, k) is a tight upper bound of the cardinality of an intersecting Sperner family or antichain of the set {1, 2,..., n}, where every collection of pairwise independent subsets is characterized by an intersection of cardinality at least k (see Theorem 1.3 in Wong and Tay).
Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - Georg Fischer, Jul 26 2023
LINKS
Eric Charles Milner, A Combinatorial Theorem On Systems of Sets, Journal of the London Mathematical Society, 43, (1968), 204-206.
W. H. W. Wong and E. G. Tay, On Cross-intersecting Sperner Families, arXiv:2001.01910 [math.CO], 2020.
EXAMPLE
The triangle T(n, k) begins
n\k| 1 2 3 4 5 6 7 8
---+-------------------------------
1 | 1
2 | 1 1
3 | 3 1 1
4 | 4 4 1 1
5 | 10 5 5 1 1
6 | 15 15 6 6 1 1
7 | 35 21 21 7 7 1 1
8 | 56 56 28 28 8 8 1 1
...
MATHEMATICA
T[n_, k_]:=Binomial[n, Floor[(n+k+1)/2]]; Table[T[n, k], {n, 12}, {k, n}]//Flatten
PROG
(PARI) T(n, k) = binomial(n, (n+k+1)\2);
vector(10, n, vector(n, k, T(n, k))) \\ Michel Marcus, Jun 01 2020
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Stefano Spezia, May 31 2020
STATUS
approved