A352525 Irregular triangle read by rows where T(n,k) is the number of integer compositions of n with k weak excedances (parts on or above the diagonal), all zeros removed.

1, 1, 2, 3, 1, 5, 3, 8, 8, 14, 17, 1, 25, 35, 4, 46, 70, 12, 87, 137, 32, 167, 268, 76, 1, 324, 525, 170, 5, 634, 1030, 367, 17, 1248, 2026, 773, 49, 2466, 3999, 1598, 129, 4887, 7914, 3267, 315, 1, 9706, 15695, 6631, 730, 6, 19308, 31181, 13393, 1631, 23

Offset: 0

Author

Gus Wiseman, Mar 22 2022

Keywords

• nonn
• tabf

Examples

			Triangle begins:
     1
     1
     2
     3     1
     5     3
     8     8
    14    17     1
    25    35     4
    46    70    12
    87   137    32
   167   268    76     1
   324   525   170     5
For example, row n = 6 counts the following compositions:
  (6)       (15)     (123)
  (51)      (24)
  (312)     (33)
  (411)     (42)
  (1113)    (114)
  (1122)    (132)
  (2112)    (141)
  (2121)    (213)
  (3111)    (222)
  (11112)   (231)
  (11121)   (321)
  (11211)   (1131)
  (21111)   (1212)
  (111111)  (1221)
            (1311)
            (2211)
            (12111)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..2489 (rows 0..200)
• MathOverflow, Why 'excedances' of permutations? [closed].

Crossrefs

Row sums are A011782.

The version for partitions is A115994.

The version for permutations is A123125, strong A173018.

Column k = 1 is A177510.

The corresponding rank statistic is A352517.

The strong opposite is A352521, first col A219282, rank statistic A352514.

The opposite version is A352522, first col A238874, rank statistic A352515.

The strong version is A352524, first column A008930, rank statistic A352516.

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

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

A352489 lists the weak excedance set of A122111.

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

Cf. A088218, A098825, A114088, A238352, A319005, A350839, A352488.

Programs

Mathematica

pdw[y_]:=Length[Select[Range[Length[y]],#<=y[[#]]&]];
DeleteCases[Table[Length[Select[Join@@ Permutations/@IntegerPartitions[n],pdw[#]==k&]],{n,0,10},{k,0,n}],0,{2}]

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); r[1]=x; [Vecrev(p) | p<-r/x]}
{ my(A=T(10)); for(i=1, #A, print(A[i])) } \\ Andrew Howroyd, Jan 19 2023