|
|
A353315
|
|
Triangle read by rows where T(n,k) is the number of integer partitions of n with k parts on or below the diagonal (weak non-excedances).
|
|
1
|
|
|
1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 0, 1, 2, 2, 3, 2, 1, 0, 1, 2, 3, 3, 3, 2, 1, 0, 1, 3, 4, 4, 4, 3, 2, 1, 0, 1, 3, 6, 5, 5, 4, 3, 2, 1, 0, 1, 4, 7, 8, 6, 6, 4, 3, 2, 1, 0, 1, 4, 9, 10, 9, 7, 6, 4, 3, 2, 1, 0, 1, 6, 10, 14, 12, 10, 8, 6, 4, 3, 2, 1, 0, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,12
|
|
LINKS
|
|
|
EXAMPLE
|
Triangle begins:
1
0 1
1 0 1
1 1 0 1
1 2 1 0 1
1 2 2 1 0 1
2 2 3 2 1 0 1
2 3 3 3 2 1 0 1
3 4 4 4 3 2 1 0 1
3 6 5 5 4 3 2 1 0 1
4 7 8 6 6 4 3 2 1 0 1
4 9 10 9 7 6 4 3 2 1 0 1
6 10 14 12 10 8 6 4 3 2 1 0 1
6 13 16 17 13 11 8 6 4 3 2 1 0 1
8 15 21 21 19 14 12 8 6 4 3 2 1 0 1
9 19 24 28 24 20 15 12 8 6 4 3 2 1 0 1
For example, row n = 9 counts the following partitions (empty column indicated by dot):
9 72 522 3222 22221 222111 2211111 21111111 . 111111111
54 81 621 4221 32211 321111 3111111
63 333 711 5211 42111 411111
432 3321 6111 51111
441 4311 33111
531
|
|
MATHEMATICA
|
pgeq[y_]:=Length[Select[Range[Length[y]], #>=y[[#]]&]];
Table[Length[Select[IntegerPartitions[n], pgeq[#]==k&]], {n, 0, 15}, {k, 0, n}]
|
|
CROSSREFS
|
The strong opposite version is A353318.
A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
A008292 is the triangle of Eulerian numbers.
A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
A238352 counts reversed partitions by fixed points, rank statistic A352822.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|