This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A353315 #8 May 16 2022 05:10:44
%S A353315 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 A353315 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 A353315 10,9,7,6,4,3,2,1,0,1,6,10,14,12,10,8,6,4,3,2,1,0,1
%N 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).
%H A353315 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A353315 Triangle begins:
%e A353315 1
%e A353315 0 1
%e A353315 1 0 1
%e A353315 1 1 0 1
%e A353315 1 2 1 0 1
%e A353315 1 2 2 1 0 1
%e A353315 2 2 3 2 1 0 1
%e A353315 2 3 3 3 2 1 0 1
%e A353315 3 4 4 4 3 2 1 0 1
%e A353315 3 6 5 5 4 3 2 1 0 1
%e A353315 4 7 8 6 6 4 3 2 1 0 1
%e A353315 4 9 10 9 7 6 4 3 2 1 0 1
%e A353315 6 10 14 12 10 8 6 4 3 2 1 0 1
%e A353315 6 13 16 17 13 11 8 6 4 3 2 1 0 1
%e A353315 8 15 21 21 19 14 12 8 6 4 3 2 1 0 1
%e A353315 9 19 24 28 24 20 15 12 8 6 4 3 2 1 0 1
%e A353315 For example, row n = 9 counts the following partitions (empty column indicated by dot):
%e A353315 9 72 522 3222 22221 222111 2211111 21111111 . 111111111
%e A353315 54 81 621 4221 32211 321111 3111111
%e A353315 63 333 711 5211 42111 411111
%e A353315 432 3321 6111 51111
%e A353315 441 4311 33111
%e A353315 531
%t A353315 pgeq[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
%t A353315 Table[Length[Select[IntegerPartitions[n],pgeq[#]==k&]],{n,0,15},{k,0,n}]
%Y A353315 Row sums are A000041.
%Y A353315 Column k = 0 is A003106.
%Y A353315 The strong version is A114088.
%Y A353315 The opposite version is A115720/A115994, rank statistic A257990.
%Y A353315 The version for permutations is A123125, strong A173018.
%Y A353315 The version for compositions is A352522, rank statistic A352515.
%Y A353315 The strong opposite version is A353318.
%Y A353315 A000700 counts self-conjugate partitions, ranked by A088902.
%Y A353315 A001522 counts partitions with a fixed point, ranked by A352827 (unproved).
%Y A353315 A008292 is the triangle of Eulerian numbers.
%Y A353315 A064428 counts partitions w/o a fixed point, ranked by A352826 (unproved).
%Y A353315 A238352 counts reversed partitions by fixed points, rank statistic A352822.
%Y A353315 A352490 gives the nonexcedance set of A122111, counted by A000701.
%Y A353315 Cf. A002620, A006918, A008290, A008930, A098825, A219282, A238874, A300788, A353319.
%K A353315 nonn,tabl
%O A353315 0,12
%A A353315 _Gus Wiseman_, May 15 2022