login
Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations and reflections.
6

%I #32 Jun 30 2018 02:18:32

%S 1,1,1,1,1,1,1,2,2,1,1,2,4,2,1,1,3,8,8,3,1,1,3,12,17,12,3,1,1,4,19,41,

%T 41,19,4,1,1,4,27,78,116,78,27,4,1,1,5,38,148,298,298,148,38,5,1,1,5,

%U 50,250,680,932,680,250,50,5,1

%N Triangle read by rows: T(n,k) is the number of k-block noncrossing partitions of n-set up to rotations and reflections.

%C Like the Narayana triangle A001263 (and unlike A152176) this triangle is symmetric.

%H Andrew Howroyd, <a href="/A209612/b209612.txt">Table of n, a(n) for n = 1..1275</a>

%H Tilman Piesk, <a href="http://en.wikiversity.org/wiki/Partition_related_number_triangles#rotref">Partition related number triangles</a>

%F T(n,k) = (A088855(n,k) + A209805(n,k))/2. - _Andrew Howroyd_, Nov 15 2017

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 2, 1;

%e 1, 2, 4, 2, 1;

%e 1, 3, 8, 8, 3, 1;

%e 1, 3, 12, 17, 12, 3, 1;

%e 1, 4, 19, 41, 41, 19, 4, 1;

%e 1, 4, 27, 78,116, 78, 27, 4, 1;

%e 1, 5, 38,148,298,298,148, 38, 5, 1

%t b[n_, k_] := Binomial[n - 1, n - k]*Binomial[n, n - k];

%t T[n_, k_] := (n*Binomial[Quotient[n - 1, 2], Quotient[k - 1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]] + DivisorSum[GCD[n, k], EulerPhi[#]* b[n/#, k/#]&] + DivisorSum[GCD[n, k - 1], EulerPhi[#]*b[n/#, (n + 1 - k)/#]&] - k*Binomial[n, k]^2/(n - k + 1))/(2*n);

%t Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jun 30 2018, after _Andrew Howroyd_ *)

%o (PARI)

%o b(n,k)=binomial(n-1,n-k)*binomial(n,n-k);

%o T(n,k)=(n*binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2) + sumdiv(gcd(n,k), d, eulerphi(d)*b(n/d,k/d)) + sumdiv(gcd(n,k-1), d, eulerphi(d)*b(n/d,(n+1-k)/d)) - k*binomial(n,k)^2/(n-k+1))/(2*n); \\ _Andrew Howroyd_, Nov 15 2017

%Y Cf. A111275 (row sums)

%Y Cf. A088855, A209805.

%K nonn,tabl

%O 1,8

%A _Tilman Piesk_, Mar 10 2012