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).

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, 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, 94, 94, 83, 58, 28, 8, 1, 0, 35, 77, 117, 162, 184, 176, 141, 86, 36, 9, 1, 0

Offset: 0

Author

Gus Wiseman, Mar 22 2022

Keywords

• nonn
• tabl

Examples

			Triangle begins:
   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  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  94  94  83  58  28   8   1   0
For example, row n = 6 counts the following compositions (empty column indicated by dot):
  (6)    (51)   (312)   (1113)   (11112)  (111111)  .
  (15)   (114)  (411)   (1122)   (11121)
  (24)   (132)  (1131)  (2112)   (11211)
  (33)   (141)  (1212)  (2121)   (21111)
  (42)   (213)  (1221)  (3111)
  (123)  (222)  (1311)  (12111)
         (231)  (2211)
         (321)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1325
• MathOverflow, Why 'excedances' of permutations? [closed].

Crossrefs

Row sums are A011782.

The version for partitions is A114088.

Row sums without the last term are A131577.

The version for permutations is A173018.

Column k = 0 is A219282.

The corresponding rank statistic is A352514.

The weak version is A352522, first column A238874, rank statistic A352515.

The opposite version is A352524, first column A008930, rank stat A352516.

The weak opposite version is A352525, first col A177510, rank stat A352517.

A008292 is the triangle of Eulerian numbers (version without zeros).

A238349 counts comps by fixed points, first col A238351, rank stat A352512.

A352490 is the strong nonexcedance set of A122111.

A352523 counts comps by nonfixed points, first A352520, rank stat A352513.

Cf. A088218, A115994, A238352, A350839, A352487, A352491.

Programs

Mathematica

pa[y_]:=Length[Select[Range[Length[y]],#>y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pa[#]==k&]],{n,0,15},{k,0,n}]

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))}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023

Extensions

Terms a(66) and beyond from Andrew Howroyd, Jan 19 2023