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%I #16 Jan 19 2023 22:35:31
%S 1,0,1,1,0,1,1,1,1,1,1,3,1,2,1,2,3,4,3,3,1,3,4,8,6,6,4,1,4,7,12,13,12,
%T 10,5,1,5,13,16,26,24,22,15,6,1,7,19,27,43,48,46,37,21,7,1,10,26,47,
%U 68,90,93,83,58,28,8,1,14,36,77,109,159,180,176,141
%N Triangle read by rows where T(n,k) is the number of integer compositions of n with k weak nonexcedances (parts on or below the diagonal).
%H Andrew Howroyd, <a href="/A352522/b352522.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%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 1 1
%e 1 3 1 2 1
%e 2 3 4 3 3 1
%e 3 4 8 6 6 4 1
%e 4 7 12 13 12 10 5 1
%e 5 13 16 26 24 22 15 6 1
%e 7 19 27 43 48 46 37 21 7 1
%e 10 26 47 68 90 93 83 58 28 8 1
%e For example, row n = 6 counts the following compositions:
%e (6) (15) (114) (123) (1113) (11112) (111111)
%e (24) (42) (132) (1311) (1122) (11121)
%e (33) (51) (141) (2112) (1131) (11211)
%e (231) (213) (2121) (1212) (12111)
%e (222) (2211) (1221)
%e (312) (3111) (21111)
%e (321)
%e (411)
%t pw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pw[#]==k&]],{n,0,15},{k,0,n}]
%o (PARI) T(n)={my(v=vector(n+1, i, i==1), r=v); for(k=1, n, v=vector(#v, j, sum(i=1, j-1, if(k>=i,x,1)*v[j-i])); r+=v); [Vecrev(p) | p<-r]}
%o { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y Row sums are A011782.
%Y The strong version for partitions is A114088.
%Y The opposite version for partitions is A115994.
%Y The version for permutations is A123125, strong A173018.
%Y Column k = 0 is A238874.
%Y The corresponding rank statistic is A352515.
%Y The strong version is A352521, first column A219282, rank statistic A352514.
%Y The strong opposite is A352524, first col A008930, rank statistic A352516.
%Y The opposite version is A352525, first col A177510, rank statistic A352517.
%Y A000041 counts integer partitions, strict A000009.
%Y A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352488 lists the weak nonexcedance set of A122111.
%Y A352523 counts comps by unfixed points, first A352520, rank stat A352513.
%Y Cf. A088218, A098825, A238352, A352489.
%K nonn,tabl
%O 0,12
%A _Gus Wiseman_, Mar 22 2022
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