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A352525 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.

%I A352525 #12 Jan 19 2023 22:35:26
%S A352525 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 A352525 324,525,170,5,634,1030,367,17,1248,2026,773,49,2466,3999,1598,129,
%U A352525 4887,7914,3267,315,1,9706,15695,6631,730,6,19308,31181,13393,1631,23
%N A352525 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 A352525 Andrew Howroyd, <a href="/A352525/b352525.txt">Table of n, a(n) for n = 0..2489</a> (rows 0..200)
%H A352525 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A352525 Triangle begins:
%e A352525      1
%e A352525      1
%e A352525      2
%e A352525      3     1
%e A352525      5     3
%e A352525      8     8
%e A352525     14    17     1
%e A352525     25    35     4
%e A352525     46    70    12
%e A352525     87   137    32
%e A352525    167   268    76     1
%e A352525    324   525   170     5
%e A352525 For example, row n = 6 counts the following compositions:
%e A352525   (6)       (15)     (123)
%e A352525   (51)      (24)
%e A352525   (312)     (33)
%e A352525   (411)     (42)
%e A352525   (1113)    (114)
%e A352525   (1122)    (132)
%e A352525   (2112)    (141)
%e A352525   (2121)    (213)
%e A352525   (3111)    (222)
%e A352525   (11112)   (231)
%e A352525   (11121)   (321)
%e A352525   (11211)   (1131)
%e A352525   (21111)   (1212)
%e A352525   (111111)  (1221)
%e A352525             (1311)
%e A352525             (2211)
%e A352525             (12111)
%t A352525 pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
%t A352525 DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n],pdw[#]==k&]],{n,0,10},{k,0,n}],0,{2}]
%o A352525 (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 A352525 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y A352525 Row sums are A011782.
%Y A352525 The version for partitions is A115994.
%Y A352525 The version for permutations is A123125, strong A173018.
%Y A352525 Column k = 1 is A177510.
%Y A352525 The corresponding rank statistic is A352517.
%Y A352525 The strong opposite is A352521, first col A219282, rank statistic A352514.
%Y A352525 The opposite version is A352522, first col A238874, rank statistic A352515.
%Y A352525 The strong version is A352524, first column A008930, rank statistic A352516.
%Y A352525 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352525 A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352525 A352489 lists the weak excedance set of A122111.
%Y A352525 A352523 counts comps by unfixed points, first A352520, rank stat A352513.
%Y A352525 Cf. A088218, A098825, A114088, A238352, A319005, A350839, A352488.
%K A352525 nonn,tabf
%O A352525 0,3
%A A352525 _Gus Wiseman_, Mar 22 2022