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A306437 Regular triangle read by rows where T(n,k) is the number of non-crossing set partitions of {1, ..., n} in which all blocks have size k.

%I A306437 #19 Feb 16 2019 18:52:15
%S A306437 1,1,1,1,0,1,1,2,0,1,1,0,0,0,1,1,5,3,0,0,1,1,0,0,0,0,0,1,1,14,0,4,0,0,
%T A306437 0,1,1,0,12,0,0,0,0,0,1,1,42,0,0,5,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,1,1,
%U A306437 132,55,22,0,6,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,1,1,429,0,0,0,0,7,0,0,0,0,0,0,1
%N A306437 Regular triangle read by rows where T(n,k) is the number of non-crossing set partitions of {1, ..., n} in which all blocks have size k.
%H A306437 Germain Kreweras, <a href="https://doi.org/10.1016/0012-365X(72)90041-6">Sur les partitions non croisées d'un cycle</a>, Discrete Math. 1 333-350 (1972).
%H A306437 Wikipedia, <a href="https://en.wikipedia.org/wiki/Noncrossing_partition">Noncrossing partition</a>.
%F A306437 If d|n, then T(n, d) = binomial(n, n/d)/(n - n/d + 1); otherwise T(n, k) = 0 [Theorem 1 of Kreweras].
%e A306437 Triangle begins:
%e A306437   1
%e A306437   1   1
%e A306437   1   0   1
%e A306437   1   2   0   1
%e A306437   1   0   0   0   1
%e A306437   1   5   3   0   0   1
%e A306437   1   0   0   0   0   0   1
%e A306437   1  14   0   4   0   0   0   1
%e A306437   1   0  12   0   0   0   0   0   1
%e A306437   1  42   0   0   5   0   0   0   0   1
%e A306437   1   0   0   0   0   0   0   0   0   0   1
%e A306437   1 132  55  22   0   6   0   0   0   0   0   1
%e A306437 Row 6 counts the following non-crossing set partitions (empty columns not shown):
%e A306437   {{1}{2}{3}{4}{5}{6}}  {{12}{34}{56}}  {{123}{456}}  {{123456}}
%e A306437                         {{12}{36}{45}}  {{126}{345}}
%e A306437                         {{14}{23}{56}}  {{156}{234}}
%e A306437                         {{16}{23}{45}}
%e A306437                         {{16}{25}{34}}
%p A306437 T:= (n, k)-> `if`(irem(n, k)=0, binomial(n, n/k)/(n-n/k+1), 0):
%p A306437 seq(seq(T(n,k), k=1..n), n=1..14);  # _Alois P. Heinz_, Feb 16 2019
%t A306437 Table[Table[If[Divisible[n,d],d/n*Binomial[n,n/d-1],0],{d,n}],{n,15}]
%Y A306437 Row sums are A194560. Column k=2 is A126120. Trisection of column k=3 is A001764.
%Y A306437 Cf. A000108, A000110, A000296, A001006, A001263, A001610, A016098, A038041, A061095, A125181, A134264.
%K A306437 nonn,tabl
%O A306437 1,8
%A A306437 _Gus Wiseman_, Feb 15 2019