A387880 Triangle read by rows where T(n,k) is the number of subdiagonal multisets summing to n (or reversed partitions of n) with maximum k.

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 1, 3, 2, 1, 0, 0, 0, 0, 1, 3, 4, 2, 0, 0, 0, 0, 0, 1, 4, 5, 4, 1, 0, 0, 0, 0, 0, 1, 4, 6, 5, 2, 0, 0, 0, 0, 0, 0, 1, 5, 8, 7, 4, 1, 0, 0, 0, 0, 0

Offset

0,18

Comments

A sequence (y_1, ..., y_k) is subdiagonal iff y_i <= i for all i = 1..k.

Links

Table of n, a(n) for n=0..77.

Example

Row n = 9 counts the following reversed partitions:
  (111111111)  (12222)     (1233)     (1134)    (11115)
               (111222)    (11133)    (1224)
               (1111122)   (11223)    (11124)
               (11111112)  (111123)   (111114)
                           (1111113)
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  1  0  0
  0  1  2  1  0  0
  0  1  2  2  0  0  0
  0  1  3  2  1  0  0  0
  0  1  3  4  2  0  0  0  0
  0  1  4  5  4  1  0  0  0  0
  0  1  4  6  5  2  0  0  0  0  0
  0  1  5  8  7  4  1  0  0  0  0  0
  0  1  5 10 10  7  2  0  0  0  0  0  0
  0  1  6 11 14 10  4  1  0  0  0  0  0  0
  0  1  6 14 16 15  7  2  0  0  0  0  0  0  0
  0  1  7 16 21 18 12  4  1  0  0  0  0  0  0  0

Mathematica

inffQ[mset_]:=And@@Table[mset[[i]]<=i, {i, Length[mset]}];
Table[Length[Select[Reverse/@IntegerPartitions[n], If[#=={}, 0, Max@@#]==k&&inffQ[#]&]], {n, 0, 15}, {k, 0, n}]

Crossrefs

Row sums are A238875.

The superdiagonal version is A388713, strictly A388724.

A000041 counts integer partitions, strict A000009.

A001522 (complement A064428), A238395 (complement A238394) count partitions w/ diagonal.

A003106 counts strictly superdiagonal partitions, strict A237979, ranks A352830.

A003114 counts superdiagonal partitions, complement A387118, strict partitions A025157.

A114088 counts partitions by excedances, see also A115720 and A115994.

A352833 counts partitions by fixed points, reverse A238352.

A388714 counts partitions with superdiagonal run-lengths, reverse A388720.

Cf. A000700, A052335, A052337, A238873, A352828, A370585, A370592, A387576, A387578, A388710, A388711.

Sequence in context: A369453 A340655 A035172 * A344858 A110174 A022909

Adjacent sequences: A387877 A387878 A387879 * A387881 A387882 A387883

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 24 2025

Status

approved