A375023 Number T(n,k) of partitions of [n] into k blocks whose element sum is <= n; triangle T(n,k), n >= 0, ceiling((n-1)/2)+signum(n) <= k <= n, read by rows.

1, 1, 1, 1, 1, 2, 1, 1, 4, 1, 5, 6, 1, 1, 15, 9, 1, 12, 32, 12, 1, 1, 58, 65, 16, 1, 31, 170, 111, 20, 1, 1, 228, 444, 186, 25, 1, 79, 914, 952, 283, 30, 1, 1, 941, 3040, 1946, 426, 36, 1, 207, 4974, 7953, 3524, 601, 42, 1, 1, 3931, 21028, 19571, 6260, 844, 49, 1

Offset

0,6

Links

Table of n, a(n) for n=0..64.

Wikipedia, Partition of a set

Example

T(6,4) = 5: 123|4|5|6, 13|24|5|6, 14|23|5|6, 15|23|4|6, 15|24|3|6.
T(7,4) = 1: 16|25|34|7.
T(7,5) = 15: 123|4|5|6|7, 124|3|5|6|7, 12|34|5|6|7, 13|24|5|6|7, 13|25|4|6|7, 14|23|5|6|7, 15|23|4|6|7, 16|23|4|5|7, 14|25|3|6|7, 15|24|3|6|7, 16|24|3|5|7, 15|2|34|6|7, 1|25|34|6|7, 16|2|34|5|7, 16|25|3|4|7.
T(7,6) = 9: 12|3|4|5|6|7, 13|2|4|5|6|7, 1|23|4|5|6|7, 14|2|3|5|6|7, 1|24|3|5|6|7, 1|2|34|5|6|7, 15|2|3|4|6|7, 1|25|3|4|6|7, 16|2|3|4|5|7.
T(7,7) = 1: 1|2|3|4|5|6|7.
Triangle T(n,k) begins:
  1;
     1;
        1;
        1, 1;
           2, 1;
           1, 4,  1;
              5,  6,  1;
              1, 15,  9,   1;
                 12, 32,  12,   1;
                  1, 58,  65,  16,   1;
                     31, 170, 111,  20,   1;
                      1, 228, 444, 186,  25,  1;
                          79, 914, 952, 283, 30, 1;
                          ...

Crossrefs

Row sums give A375099.

T(n+1,n) gives A002620.

Sequence in context: A112987 A125138 A372647 * A021477 A124939 A187800

Adjacent sequences: A375020 A375021 A375022 * A375024 A375025 A375026

Keyword

nonn,tabf

Author

Alois P. Heinz, Aug 01 2024

Status

approved