A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 10, 5, 5, 1, 1, 15, 15, 6, 6, 1, 1, 35, 21, 21, 7, 7, 1, 1, 56, 56, 28, 28, 8, 8, 1, 1, 126, 84, 84, 36, 36, 9, 9, 1, 1, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1, 792, 792, 495, 495, 220, 220, 66, 66, 12, 12, 1, 1

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

Table of n, a(n) for n=1..78.

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.

Index entries for sequences related to antichains.

Formula

T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.

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

Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.

Cf. A037951 (k=3), A037952 (k=1), A037953 (k=5), A037954 (k=7), A037955 (k=2), A037956 (k=4), A037957 (k=6), A037958 (k=8), A045621 (row sums).

Sequence in context: A102716 A173076 A134510 * A171145 A271644 A262191

Adjacent sequences: A335319 A335320 A335321 * A335323 A335324 A335325

Keyword

nonn,tabl

Author

Stefano Spezia, May 31 2020

Status

approved