A352523 Number of integer compositions of n with exactly k nonfixed points (parts not on the diagonal).

1, 1, 0, 0, 2, 0, 1, 1, 2, 0, 0, 4, 2, 2, 0, 0, 5, 5, 4, 2, 0, 1, 3, 12, 8, 6, 2, 0, 0, 7, 14, 19, 14, 8, 2, 0, 0, 8, 21, 33, 32, 22, 10, 2, 0, 0, 9, 30, 54, 63, 54, 32, 12, 2, 0, 1, 6, 47, 80, 116, 116, 86, 44, 14, 2, 0, 0, 11, 53, 129, 194, 229, 202, 130, 58, 16, 2, 0

Offset: 0

Author

Gus Wiseman, Mar 26 2022

Keywords

• nonn
• tabl

Comments

A nonfixed point in a composition c is an index i such that c_i != i.

Examples

			Triangle begins:
   1
   1   0
   0   2   0
   1   1   2   0
   0   4   2   2   0
   0   5   5   4   2   0
   1   3  12   8   6   2   0
   0   7  14  19  14   8   2   0
   0   8  21  33  32  22  10   2   0
   0   9  30  54  63  54  32  12   2   0
   1   6  47  80 116 116  86  44  14   2   0
   ...
For example, row n = 6 counts the following compositions (empty column indicated by dot):
  (123)  (6)   (24)    (231)    (2112)   (21111)    .
         (15)  (33)    (312)    (2121)   (111111)
         (42)  (51)    (411)    (3111)
               (114)   (1113)   (11112)
               (132)   (1122)   (11121)
               (141)   (1311)   (11211)
               (213)   (2211)
               (222)   (12111)
               (321)
               (1131)
               (1212)
               (1221)
		

Links

• John Tyler Rascoe, Rows n = 0..130, flattened

Crossrefs

Column k = 0 is A010054.

Row sums are A011782.

The version for permutations is A098825.

The corresponding rank statistic is A352513.

Column k = 1 is A352520.

A238349 and A238350 count comps by fixed points, first col A238351, rank stat A352512.

A352486 gives the nonfixed points of A122111, counted by A330644.

A352521 counts comps by strong nonexcedances, first A219282, stat A352514.

A352522 counts comps by weak nonexcedances, first col A238874, stat A352515.

A352524 counts comps by strong excedances, first col A008930, stat A352516.

A352525 counts comps by weak excedances, first col A177510, stat A352517.

Cf. A000700, A064428, A088218, A114088, A115994, A123125, A173018, A238352.

Programs

Maple

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      add(`if`(i=j, 1, x)*b(n-j, i+1), j=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 19 2025

Mathematica

pnq[y_]:=Length[Select[Range[Length[y]],#!=y[[#]]&]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],pnq[#]==k&]],{n,0,9},{k,0,n}]

PARI

T_xy(max_row) = {my(N=max_row+1, x='x+O('x^N), h= sum(i=0, N, prod(j=1, i, y*(x/(1-x)-x^j)+x^j))); vector(N, n, my(r=Vecrev(polcoeff(h, n-1))); if(n<2, r, concat(r,[0])))}
T_xy(10) \\ John Tyler Rascoe, Mar 21 2025

Formula

G.f.: Sum_{i>=0} Product_{j=1..i} y*(x/(1-x) - x^j) + x^j. - John Tyler Rascoe, Mar 19 2025