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A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.

%I A352524 #17 Jan 02 2023 21:55:12
%S A352524 1,1,1,1,2,2,3,5,6,9,1,11,18,3,21,35,8,41,67,20,80,131,44,1,157,257,
%T A352524 94,4,310,505,197,12,614,996,406,32,1218,1973,825,80,2421,3915,1669,
%U A352524 186,1,4819,7781,3364,415,5,9602,15486,6762,901,17,19147,30855,13567,1918,49
%N A352524 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k excedances (parts above the diagonal), all zeros removed.
%H A352524 Andrew Howroyd, <a href="/A352524/b352524.txt">Table of n, a(n) for n = 0..2507</a> (rows 0..200)
%H A352524 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A352524 Triangle begins:
%e A352524      1
%e A352524      1
%e A352524      1     1
%e A352524      2     2
%e A352524      3     5
%e A352524      6     9     1
%e A352524     11    18     3
%e A352524     21    35     8
%e A352524     41    67    20
%e A352524     80   131    44     1
%e A352524    157   257    94     4
%e A352524    310   505   197    12
%e A352524    614   996   406    32
%e A352524 For example, row n = 5 counts the following compositions:
%e A352524   (113)    (5)     (23)
%e A352524   (122)    (14)
%e A352524   (1112)   (32)
%e A352524   (1121)   (41)
%e A352524   (1211)   (131)
%e A352524   (11111)  (212)
%e A352524            (221)
%e A352524            (311)
%e A352524            (2111)
%t A352524 pd[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t A352524 DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n],pd[#]==k&]],{n,0,10},{k,0,n}],0,{2}]
%o A352524 (PARI)
%o A352524 S(v,u)={vector(#v, k, sum(i=1, k-1, v[k-i]*u[i]))}
%o A352524 T(n)={my(v=vector(1+n), s); v[1]=1; s=v; for(i=1, n, v=S(v, vector(n, j, if(j>i,'x,1))); s+=v); [Vecrev(p) | p<-s]}
%o A352524 { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 02 2023
%Y A352524 The version for permutations is A008292, weak A123125.
%Y A352524 Column k = 0 is A008930.
%Y A352524 Row sums are A011782.
%Y A352524 The opposite version for partitions is A114088.
%Y A352524 The weak version for partitions is A115994.
%Y A352524 Column k = 1 is A351983.
%Y A352524 The corresponding rank statistic is A352516.
%Y A352524 The opposite version is A352521, first col A219282, rank statistic A352514.
%Y A352524 The weak opposite version is A352522, first col A238874, rank stat A352515.
%Y A352524 The weak version is A352525, first col (k = 1) A177510, rank stat A352517.
%Y A352524 A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352524 A352487 lists the excedance set of A122111, opposite A352490.
%Y A352524 A352523 counts comps by unfixed points, first A352520, rank stat A352513.
%Y A352524 Cf. A088218, A098825, A238352, A350839.
%K A352524 nonn,tabf
%O A352524 0,5
%A A352524 _Gus Wiseman_, Mar 22 2022