A374932 - OEIS

%I #31 Aug 02 2024 17:41:00

%S 1,1,1,1,2,1,1,1,3,3,3,3,1,1,1,6,6,8,9,9,5,3,3,1,1,1,12,14,20,31,26,

%T 32,19,14,11,10,5,3,3,1,1,1,26,31,59,78,111,108,113,76,67,57,39,39,21,

%U 16,12,10,5,3,3,1,1,1,57,84,140,260,321,458,427,500,326,300,284,229,182,159,107,79,64,46,41,23,17,12,10,5,3,3,1,1,1

%N Number T(n,k) of partitions of [n] such that the maximal block element sum equals k; triangle T(n,k), n>=0, n <= k <= A000217(n), read by rows.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%e T(5,7) = 8: 124|3|5, 12|34|5, 13|25|4, 14|25|3, 15|2|34, 1|25|34, 1|2|34|5, 1|25|3|4.

%e T(6,6) = 12: 123|4|5|6, 12|3|4|5|6, 13|24|5|6, 13|2|4|5|6, 14|23|5|6, 15|23|4|6, 1|23|4|5|6, 14|2|3|5|6, 15|24|3|6, 1|24|3|5|6, 15|2|3|4|6, 1|2|3|4|5|6.

%e T(6,7) = 14: 124|3|5|6, 12|34|5|6, 13|25|4|6, 16|23|4|5, 14|25|3|6, 16|24|3|5, 15|2|34|6, 16|25|34, 1|25|34|6, 16|2|34|5, 1|2|34|5|6, 16|25|3|4, 1|25|3|4|6, 16|2|3|4|5.

%e Triangle T(n,k) begins:

%e   1;

%e      1;

%e         1, 1;

%e            2, 1, 1,  1;

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

%e                  6,  6,  8,  9,  9,  5,  3,  3,  1,  1,  1;

%e                     12, 14, 20, 31, 26, 32, 19, 14, 11, 10, 5, 3, 3, 1, 1, 1;

%e                     ...

%Y Row sums give A000110.

%Y Main diagonal gives A375099.

%Y Number of terms in row n is A000124(n-1) for n>=1.

%Y Reversed rows converge to A294617.

%Y Cf. A000217.

%K nonn,tabf

%O 0,5

%A _Alois P. Heinz_, Aug 01 2024