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A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A370366 #21 Feb 17 2024 17:30:57
%S A370366 1,1,0,1,0,0,1,0,0,0,1,0,2,0,0,1,0,9,8,0,0,1,0,34,252,60,0,0,1,0,125,
%T A370366 5672,14337,544,0,0,1,0,461,125750,2604732,1327104,6040,0,0,1,0,1715,
%U A370366 2857472,488360625,2533087904,182407545,79008,0,0
%N A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A370366 Alois P. Heinz, <a href="/A370366/b370366.txt">Antidiagonals n = 0..54, flattened</a>
%H A370366 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A370366 A(n,k) = A060540(n,k) - A370363(n,k) for n,k >= 1.
%e A370366 A(2,3) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
%e A370366 Square array A(n,k) begins:
%e A370366   1, 1,   1,       1,          1,             1, ...
%e A370366   0, 0,   0,       0,          0,             0, ...
%e A370366   0, 0,   2,       9,         34,           125, ...
%e A370366   0, 0,   8,     252,       5672,        125750, ...
%e A370366   0, 0,  60,   14337,    2604732,     488360625, ...
%e A370366   0, 0, 544, 1327104, 2533087904, 5192229797500, ...
%p A370366 A:= proc(n, k) `if`(k=0,`if`(n=0, 1, 0), add(
%p A370366       (-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
%p A370366     end:
%p A370366 seq(seq(A(n, d-n), n=0..d), d=0..10);
%Y A370366 Columns k=0+1,2-3 give: A000007, A053871, A370357.
%Y A370366 Rows n=0-2 give: A000012, A000004, A010763(n-1) for k>0.
%Y A370366 Main diagonal gives A370367.
%Y A370366 Antidiagonal sums give A370368.
%Y A370366 Cf. A060540, A370363.
%K A370366 nonn,tabl
%O A370366 0,13
%A A370366 _Alois P. Heinz_, Feb 16 2024