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A352522 Triangle read by rows where T(n,k) is the number of integer compositions of n with k weak nonexcedances (parts on or below the diagonal).

%I A352522 #16 Jan 19 2023 22:35:31
%S A352522 1,0,1,1,0,1,1,1,1,1,1,3,1,2,1,2,3,4,3,3,1,3,4,8,6,6,4,1,4,7,12,13,12,
%T A352522 10,5,1,5,13,16,26,24,22,15,6,1,7,19,27,43,48,46,37,21,7,1,10,26,47,
%U A352522 68,90,93,83,58,28,8,1,14,36,77,109,159,180,176,141
%N A352522 Triangle read by rows where T(n,k) is the number of integer compositions of n with k weak nonexcedances (parts on or below the diagonal).
%H A352522 Andrew Howroyd, <a href="/A352522/b352522.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A352522 MathOverflow, <a href="https://mathoverflow.net/questions/359684/why-excedances-of-permutations">Why 'excedances' of permutations? [closed]</a>.
%e A352522 Triangle begins:
%e A352522    1
%e A352522    0   1
%e A352522    1   0   1
%e A352522    1   1   1   1
%e A352522    1   3   1   2   1
%e A352522    2   3   4   3   3   1
%e A352522    3   4   8   6   6   4   1
%e A352522    4   7  12  13  12  10   5   1
%e A352522    5  13  16  26  24  22  15   6   1
%e A352522    7  19  27  43  48  46  37  21   7   1
%e A352522   10  26  47  68  90  93  83  58  28   8   1
%e A352522 For example, row n = 6 counts the following compositions:
%e A352522   (6)   (15)   (114)  (123)   (1113)   (11112)  (111111)
%e A352522   (24)  (42)   (132)  (1311)  (1122)   (11121)
%e A352522   (33)  (51)   (141)  (2112)  (1131)   (11211)
%e A352522         (231)  (213)  (2121)  (1212)   (12111)
%e A352522                (222)  (2211)  (1221)
%e A352522                (312)  (3111)  (21111)
%e A352522                (321)
%e A352522                (411)
%t A352522 pw[y_]:=Length[Select[Range[Length[y]],#>=y[[#]]&]];
%t A352522 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pw[#]==k&]],{n,0,15},{k,0,n}]
%o A352522 (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); [Vecrev(p) | p<-r]}
%o A352522 { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 19 2023
%Y A352522 Row sums are A011782.
%Y A352522 The strong version for partitions is A114088.
%Y A352522 The opposite version for partitions is A115994.
%Y A352522 The version for permutations is A123125, strong A173018.
%Y A352522 Column k = 0 is A238874.
%Y A352522 The corresponding rank statistic is A352515.
%Y A352522 The strong version is A352521, first column A219282, rank statistic A352514.
%Y A352522 The strong opposite is A352524, first col A008930, rank statistic A352516.
%Y A352522 The opposite version is A352525, first col A177510, rank statistic A352517.
%Y A352522 A000041 counts integer partitions, strict A000009.
%Y A352522 A008292 is the triangle of Eulerian numbers (version without zeros).
%Y A352522 A238349 counts comps by fixed points, first col A238351, rank stat A352512.
%Y A352522 A352488 lists the weak nonexcedance set of A122111.
%Y A352522 A352523 counts comps by unfixed points, first A352520, rank stat A352513.
%Y A352522 Cf. A088218, A098825, A238352, A352489.
%K A352522 nonn,tabl
%O A352522 0,12
%A A352522 _Gus Wiseman_, Mar 22 2022