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%I #17 Jan 02 2023 21:55:12
%S 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 94,4,310,505,197,12,614,996,406,32,1218,1973,825,80,2421,3915,1669,
%U 186,1,4819,7781,3364,415,5,9602,15486,6762,901,17,19147,30855,13567,1918,49
%N 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 Andrew Howroyd, <a href="/A352524/b352524.txt">Table of n, a(n) for n = 0..2507</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 1 1
%e 2 2
%e 3 5
%e 6 9 1
%e 11 18 3
%e 21 35 8
%e 41 67 20
%e 80 131 44 1
%e 157 257 94 4
%e 310 505 197 12
%e 614 996 406 32
%e For example, row n = 5 counts the following compositions:
%e (113) (5) (23)
%e (122) (14)
%e (1112) (32)
%e (1121) (41)
%e (1211) (131)
%e (11111) (212)
%e (221)
%e (311)
%e (2111)
%t pd[y_]:=Length[Select[Range[Length[y]],#<y[[#]]&]];
%t DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n],pd[#]==k&]],{n,0,10},{k,0,n}],0,{2}]
%o (PARI)
%o S(v,u)={vector(#v, k, sum(i=1, k-1, v[k-i]*u[i]))}
%o 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 { my(A=T(12)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 02 2023
%Y The version for permutations is A008292, weak A123125.
%Y Column k = 0 is A008930.
%Y Row sums are A011782.
%Y The opposite version for partitions is A114088.
%Y The weak version for partitions is A115994.
%Y Column k = 1 is A351983.
%Y The corresponding rank statistic is A352516.
%Y The opposite version is A352521, first col A219282, rank statistic A352514.
%Y The weak opposite version is A352522, first col A238874, rank stat A352515.
%Y The weak version is A352525, first col (k = 1) A177510, rank stat A352517.
%Y A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352487 lists the excedance set of A122111, opposite A352490.
%Y A352523 counts comps by unfixed points, first A352520, rank stat A352513.
%Y Cf. A088218, A098825, A238352, A350839.
%K nonn,tabf
%O 0,5
%A _Gus Wiseman_, Mar 22 2022
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