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A352521 Triangle read by rows where T(n,k) is the number of integer compositions of n with k strong nonexcedances (parts below the diagonal).

%I A352521 #16 Jan 19 2023 22:35:35
%S A352521 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 A352521 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 A352521 94,94,83,58,28,8,1,0,35,77,117,162,184,176,141,86,36,9,1,0
%N A352521 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 A352521 Andrew Howroyd, <a href="/A352521/b352521.txt">Table of n, a(n) for n = 0..1325</a>
%H A352521 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A352521 Triangle begins:
%e A352521    1
%e A352521    1   0
%e A352521    1   1   0
%e A352521    2   1   1   0
%e A352521    3   2   2   1   0
%e A352521    4   5   3   3   1   0
%e A352521    6   8   7   6   4   1   0
%e A352521    9  12  15  12  10   5   1   0
%e A352521   13  19  27  25  22  15   6   1   0
%e A352521   18  32  43  51  46  37  21   7   1   0
%e A352521   25  51  70  94  94  83  58  28   8   1   0
%e A352521 For example, row n = 6 counts the following compositions (empty column indicated by dot):
%e A352521   (6)    (51)   (312)   (1113)   (11112)  (111111)  .
%e A352521   (15)   (114)  (411)   (1122)   (11121)
%e A352521   (24)   (132)  (1131)  (2112)   (11211)
%e A352521   (33)   (141)  (1212)  (2121)   (21111)
%e A352521   (42)   (213)  (1221)  (3111)
%e A352521   (123)  (222)  (1311)  (12111)
%e A352521          (231)  (2211)
%e A352521          (321)
%t A352521 pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
%t A352521 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pa[#]==k&]],{n,0,15},{k,0,n}]
%o A352521 (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 A352521 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y A352521 Row sums are A011782.
%Y A352521 The version for partitions is A114088.
%Y A352521 Row sums without the last term are A131577.
%Y A352521 The version for permutations is A173018.
%Y A352521 Column k = 0 is A219282.
%Y A352521 The corresponding rank statistic is A352514.
%Y A352521 The weak version is A352522, first column A238874, rank statistic A352515.
%Y A352521 The opposite version is A352524, first column A008930, rank stat A352516.
%Y A352521 The weak opposite version is A352525, first col A177510, rank stat A352517.
%Y A352521 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352521 A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352521 A352490 is the strong nonexcedance set of A122111.
%Y A352521 A352523 counts comps by nonfixed points, first A352520, rank stat A352513.
%Y A352521 Cf. A088218, A115994, A238352, A350839, A352487, A352491.
%K A352521 nonn,tabl
%O A352521 0,7
%A A352521 _Gus Wiseman_, Mar 22 2022
%E A352521 Terms a(66) and beyond from _Andrew Howroyd_, Jan 19 2023