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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.
20

%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