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A325982
Triangle read by rows: T(n, k) = binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1, with n >= 1 and 0 <= k < n/2.
3
1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 3, 13, 1, 1, 3, 16, 1, 1, 3, 19, 53, 1, 1, 3, 22, 75, 1, 1, 3, 25, 101, 206, 1, 1, 3, 28, 131, 316, 1, 1, 3, 31, 165, 461, 787, 1, 1, 3, 34, 203, 646, 1267, 1, 1, 3, 37, 245, 876, 1947, 2997, 1, 1, 3, 40, 291, 1156, 2878, 4978
OFFSET
1,9
COMMENTS
Given X an n-element set and F a family of k-subsets of X. If n > 2*k and F is a nontrivial intersecting family, then the cardinality of F is almost equal to T(n, k). A family F is called trivial if all its members contain a fixed element of X (see Hilton-Milner Theorem in Links).
LINKS
Peter Frankl, A simple proof of the Hilton-Milner theorem, Moscow Journal of Combinatorics and Number Theory, Volume 8, Number 2 (2019), 97-101.
Peter Frankl and Zoltán Füredi, Non-trivial Intersecting Families, Journal of Combinatorial Theory, Series A, Vol. 41, No. 1, January 1986.
Peter Frankl and Andrey Kupavskii, Sharp results concerning disjoint cross-intersecting families, arXiv:1905.08123 [math.CO], 2019.
Peter Frankl and Andrey Kupavskii, Uniform intersecting families with large covering number, arXiv:2106.05344 [math.CO], 2021. See p. 2.
Anthony J. W. Hilton and Eric Charles Milner, Some Intersection Theorems for Systems of Finite Sets, The Quarterly Journal of Mathematics, Volume 18, Issue 1, 1967, Pages 369-384.
Russ Woodroofe, An algebraic groups perspective on Erdős-Ko-Rado, arXiv:2007.03707 [math.CO], 2020. See p. 2.
FORMULA
T(n, k) = A007318(n - 1, k - 1) - A007318(n - k - 1, k - 1) + 1.
EXAMPLE
The triangle T(n, k) begins
n\k| 0 1 2 3 4
---+----------------------
1 | 1
2 | 1
3 | 1 1
4 | 1 1
5 | 1 1 3
6 | 1 1 3
7 | 1 1 3 13
8 | 1 1 3 16
9 | 1 1 3 19 53
10 | 1 1 3 22 75
...
MAPLE
a := (n, k) -> binomial(n-1, k-1)-binomial(n-k-1, k-1)+1: seq(seq(a(n, k), k = 0 .. floor((n-1)/2)), n = 1 .. 15);
MATHEMATICA
T[n_, k_]:=Binomial[n-1, k-1]-Binomial[n-k-1, k-1]+1; Flatten[Table[T[n, k], {n, 1, 15}, {k, 0, Floor[(n-1)/2]}]]
PROG
(GAP) Flat(List([1..15], n->List([0..Int((n-1)/2)], k->Binomial(n-1, k-1)-Binomial(n-k-1, k-1)+1)));
(Magma) [[Binomial(n-1, k-1)-Binomial(n-k-1, k-1)+1: k in [0..Floor((n-1)/2)]]: n in [1 .. 15]]; // triangle output
(PARI)
T(n, k) = binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1;
tabf(nn) = for(i=1, nn, for(j=0, floor((i-1)/2), print1(T(i, j), ", ")); print);
tabf(15) \\ triangle output
CROSSREFS
Cf. A004526, A007318, A325983 (row sums).
Sequence in context: A046929 A068695 A110787 * A202338 A139002 A373438
KEYWORD
nonn,tabf
AUTHOR
Stefano Spezia, May 29 2019
STATUS
approved