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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

%I #8 May 16 2022 05:10:44

%S 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,

%T 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,

%U 10,9,7,6,4,3,2,1,0,1,6,10,14,12,10,8,6,4,3,2,1,0,1

%N 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).

%H MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.

%e Triangle begins:

%e 1

%e 0 1

%e 1 0 1

%e 1 1 0 1

%e 1 2 1 0 1

%e 1 2 2 1 0 1

%e 2 2 3 2 1 0 1

%e 2 3 3 3 2 1 0 1

%e 3 4 4 4 3 2 1 0 1

%e 3 6 5 5 4 3 2 1 0 1

%e 4 7 8 6 6 4 3 2 1 0 1

%e 4 9 10 9 7 6 4 3 2 1 0 1

%e 6 10 14 12 10 8 6 4 3 2 1 0 1

%e 6 13 16 17 13 11 8 6 4 3 2 1 0 1

%e 8 15 21 21 19 14 12 8 6 4 3 2 1 0 1

%e 9 19 24 28 24 20 15 12 8 6 4 3 2 1 0 1

%e For example, row n = 9 counts the following partitions (empty column indicated by dot):

%e 9 72 522 3222 22221 222111 2211111 21111111 . 111111111

%e 54 81 621 4221 32211 321111 3111111

%e 63 333 711 5211 42111 411111

%e 432 3321 6111 51111

%e 441 4311 33111

%e 531

%t pgeq[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];

%t Table[Length[Select[IntegerPartitions[n],pgeq[#]==k&]],{n,0,15},{k,0,n}]

%Y Row sums are A000041.

%Y Column k = 0 is A003106.

%Y The strong version is A114088.

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

%Y The version for permutations is A123125, strong A173018.

%Y The version for compositions is A352522, rank statistic A352515.

%Y The strong opposite version is A353318.

%Y A000700 counts self-conjugate partitions, ranked by A088902.

%Y A001522 counts partitions with a fixed point, ranked by A352827 (unproved).

%Y A008292 is the triangle of Eulerian numbers.

%Y A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).

%Y A238352 counts reversed partitions by fixed points, rank statistic A352822.

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

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

%K nonn,tabl

%O 0,12

%A _Gus Wiseman_, May 15 2022