A221879 Triangle T(n,k) read by rows: Number of order-reversing full contraction mappings (of an n-chain) with 1 fixed point and height exactly k.

1, 2, 0, 3, 2, 1, 4, 6, 4, 0, 5, 12, 12, 4, 1, 6, 20, 28, 18, 6, 0, 7, 30, 55, 52, 27, 6, 1, 8, 42, 96, 120, 88, 36, 8, 0, 9, 56, 154, 240, 230, 136, 48, 8, 1, 10, 72, 232, 434, 516, 400, 200, 60, 10, 0, 11, 90, 333, 728, 1036, 996, 650, 280, 75, 10, 1

Offset

1,2

Comments

Row sums are A059570.

Links

Table of n, a(n) for n=1..66.

A. D. Adeshola, V. Maltcev and A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, arXiv:1303.7428 [math.CO], 2013.

A. D. Adeshola, A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, JMCC 106 (2017) 37-49

Formula

T(n, 1) = 1, T(2,2) = 0 and T(n,k) = (n-k+1)*C(n-2,k-1) + T(n-2,k-2) for k > 0.

Sum_{k=1..n} T(n,k) = A059570(n).

Example

T (4,6) = 6 because there are exactly 6 order-reversing full contraction mappings (of a 4-chain) with 1 fixed point and of height exactly 2, namely: (3222), (2221), (2211), (4433), (4333), (3332).
Triangle starts:
  1,
  2, 0,
  3, 2, 1,
  4, 6, 4, 0,
  5, 12, 12, 4, 1,
  6, 20, 28, 18, 6, 0,
  7, 30, 55, 52, 27, 6, 1,
  8, 42, 96, 120, 88, 36, 8, 0,
  9, 56, 154, 240, 230, 136, 48, 8, 1,
  10, 72, 232, 434, 516, 400, 200, 60, 10, 0,
  11, 90, 333, 728, 1036, 996, 650, 280, 75, 10, 1
  ...

Maple

A221879 := proc(n, k)
    option remember ;
    if n<1 then
        0 ;
    elif n=1 then
        if k = 1 then
            1;
        else
            0 ;
        end if;
    else
        if n = 2 and k=2 then
            0;
        else
            (n-k+1)*binomial(n-2, k-1)+procname(n-2, k-2) ;
        end if;
    end if;
end proc:
seq(seq( A221879(n, k), k=1..n), n=1..20) ; # R. J. Mathar, Aug 15 2025

Crossrefs

Cf. A059570 (row sums), A221876, A221877, A221878, A221880, A221881, A221882.

Sequence in context: A241556 A242029 A090722 * A171934 A303205 A082785

Adjacent sequences: A221876 A221877 A221878 * A221880 A221881 A221882

Keyword

nonn,tabl,easy

Author

Abdullahi Umar, Feb 28 2013

Status

approved