A325982 - OEIS

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.

%I #39 Sep 08 2022 08:46:24

%S 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,

%T 1,3,25,101,206,1,1,3,28,131,316,1,1,3,31,165,461,787,1,1,3,34,203,

%U 646,1267,1,1,3,37,245,876,1947,2997,1,1,3,40,291,1156,2878,4978

%N 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.

%C 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).

%H Stefano Spezia, <a href="/A325982/b325982.txt">First 200 rows of the triangle, flattened</a>

%H Peter Frankl, <a href="https://dx.doi.org/10.2140/moscow.2019.8.97">A simple proof of the Hilton-Milner theorem</a>, Moscow Journal of Combinatorics and Number Theory, Volume 8, Number 2 (2019), 97-101.

%H Peter Frankl and Zoltán Füredi, <a href="https://doi.org/10.1016/0097-3165(86)90121-4">Non-trivial Intersecting Families</a>, Journal of Combinatorial Theory, Series A, Vol. 41, No. 1, January 1986.

%H Peter Frankl and Andrey Kupavskii, <a href="https://arxiv.org/abs/1905.08123">Sharp results concerning disjoint cross-intersecting families</a>, arXiv:1905.08123 [math.CO], 2019.

%H Peter Frankl and Andrey Kupavskii, <a href="https://arxiv.org/abs/2106.05344">Uniform intersecting families with large covering number</a>, arXiv:2106.05344 [math.CO], 2021. See p. 2.

%H Anthony J. W. Hilton and Eric Charles Milner, <a href="https://doi.org/10.1093/qmath/18.1.369">Some Intersection Theorems for Systems of Finite Sets</a>, The Quarterly Journal of Mathematics, Volume 18, Issue 1, 1967, Pages 369-384.

%H Russ Woodroofe, <a href="https://arxiv.org/abs/2007.03707">An algebraic groups perspective on Erdős-Ko-Rado</a>, arXiv:2007.03707 [math.CO], 2020. See p. 2.

%F T(n, k) = A007318(n - 1, k - 1) - A007318(n - k - 1, k - 1) + 1.

%e The triangle T(n, k) begins

%e n\k| 0 1 2 3 4

%e ---+----------------------

%e 1 | 1

%e 2 | 1

%e 3 | 1 1

%e 4 | 1 1

%e 5 | 1 1 3

%e 6 | 1 1 3

%e 7 | 1 1 3 13

%e 8 | 1 1 3 16

%e 9 | 1 1 3 19 53

%e 10 | 1 1 3 22 75

%e ...

%p 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);

%t 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]}]]

%o (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)));

%o (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

%o (PARI)

%o T(n, k) = binomial(n - 1, k - 1) - binomial(n - k - 1, k - 1) + 1;

%o tabf(nn) = for(i=1, nn, for(j=0, floor((i-1)/2), print1(T(i, j), ", ")); print);

%o tabf(15) \\ triangle output

%Y Cf. A004526, A007318, A325983 (row sums).

%K nonn,tabf

%O 1,9

%A _Stefano Spezia_, May 29 2019