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%I #12 Jan 19 2023 22:35:26
%S 1,1,2,3,1,5,3,8,8,14,17,1,25,35,4,46,70,12,87,137,32,167,268,76,1,
%T 324,525,170,5,634,1030,367,17,1248,2026,773,49,2466,3999,1598,129,
%U 4887,7914,3267,315,1,9706,15695,6631,730,6,19308,31181,13393,1631,23
%N Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k weak excedances (parts on or above the diagonal), all zeros removed.
%H Andrew Howroyd, <a href="/A352525/b352525.txt">Table of n, a(n) for n = 0..2489</a> (rows 0..200)
%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 1
%e 2
%e 3 1
%e 5 3
%e 8 8
%e 14 17 1
%e 25 35 4
%e 46 70 12
%e 87 137 32
%e 167 268 76 1
%e 324 525 170 5
%e For example, row n = 6 counts the following compositions:
%e (6) (15) (123)
%e (51) (24)
%e (312) (33)
%e (411) (42)
%e (1113) (114)
%e (1122) (132)
%e (2112) (141)
%e (2121) (213)
%e (3111) (222)
%e (11112) (231)
%e (11121) (321)
%e (11211) (1131)
%e (21111) (1212)
%e (111111) (1221)
%e (1311)
%e (2211)
%e (12111)
%t pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
%t DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n],pdw[#]==k&]],{n,0,10},{k,0,n}],0,{2}]
%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); r[1]=x; [Vecrev(p) | p<-r/x]}
%o { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y Row sums are A011782.
%Y The version for partitions is A115994.
%Y The version for permutations is A123125, strong A173018.
%Y Column k = 1 is A177510.
%Y The corresponding rank statistic is A352517.
%Y The strong opposite is A352521, first col A219282, rank statistic A352514.
%Y The opposite version is A352522, first col A238874, rank statistic A352515.
%Y The strong version is A352524, first column A008930, rank statistic A352516.
%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 A352489 lists the weak excedance set of A122111.
%Y A352523 counts comps by unfixed points, first A352520, rank stat A352513.
%Y Cf. A088218, A098825, A114088, A238352, A319005, A350839, A352488.
%K nonn,tabf
%O 0,3
%A _Gus Wiseman_, Mar 22 2022
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