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%I #16 Jan 19 2023 22:35:35
%S 1,1,0,1,1,0,2,1,1,0,3,2,2,1,0,4,5,3,3,1,0,6,8,7,6,4,1,0,9,12,15,12,
%T 10,5,1,0,13,19,27,25,22,15,6,1,0,18,32,43,51,46,37,21,7,1,0,25,51,70,
%U 94,94,83,58,28,8,1,0,35,77,117,162,184,176,141,86,36,9,1,0
%N Triangle read by rows where T(n,k) is the number of integer compositions of n with k strong nonexcedances (parts below the diagonal).
%H Andrew Howroyd, <a href="/A352521/b352521.txt">Table of n, a(n) for n = 0..1325</a>
%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 0
%e 1 1 0
%e 2 1 1 0
%e 3 2 2 1 0
%e 4 5 3 3 1 0
%e 6 8 7 6 4 1 0
%e 9 12 15 12 10 5 1 0
%e 13 19 27 25 22 15 6 1 0
%e 18 32 43 51 46 37 21 7 1 0
%e 25 51 70 94 94 83 58 28 8 1 0
%e For example, row n = 6 counts the following compositions (empty column indicated by dot):
%e (6) (51) (312) (1113) (11112) (111111) .
%e (15) (114) (411) (1122) (11121)
%e (24) (132) (1131) (2112) (11211)
%e (33) (141) (1212) (2121) (21111)
%e (42) (213) (1221) (3111)
%e (123) (222) (1311) (12111)
%e (231) (2211)
%e (321)
%t pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pa[#]==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); vector(#v, i, Vecrev(r[i], i))}
%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 A114088.
%Y Row sums without the last term are A131577.
%Y The version for permutations is A173018.
%Y Column k = 0 is A219282.
%Y The corresponding rank statistic is A352514.
%Y The weak version is A352522, first column A238874, rank statistic A352515.
%Y The opposite version is A352524, first column A008930, rank stat A352516.
%Y The weak opposite version is A352525, first col A177510, rank stat A352517.
%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 A352490 is the strong nonexcedance set of A122111.
%Y A352523 counts comps by nonfixed points, first A352520, rank stat A352513.
%Y Cf. A088218, A115994, A238352, A350839, A352487, A352491.
%K nonn,tabl
%O 0,7
%A _Gus Wiseman_, Mar 22 2022
%E Terms a(66) and beyond from _Andrew Howroyd_, Jan 19 2023
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