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, 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

Offset: 0

Author

Gus Wiseman, May 15 2022

Keywords

• nonn
• tabl

Examples

			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
		

Links

• MathOverflow, Why 'excedances' of permutations? [closed].

Crossrefs

Row sums are A000041.

Column k = 0 is A003106.

The strong version is A114088.

The opposite version is A115720/A115994, rank statistic A257990.

The version for permutations is A123125, strong A173018.

The version for compositions is A352522, rank statistic A352515.

The strong opposite version is A353318.

A000700 counts self-conjugate partitions, ranked by A088902.

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.

A352490 gives the nonexcedance set of A122111, counted by A000701.

Cf. A002620, A006918, A008290, A008930, A098825, A219282, A238874, A300788, A353319.

Programs

Mathematica

pgeq[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
Table[Length[Select[IntegerPartitions[n],pgeq[#]==k&]],{n,0,15},{k,0,n}]