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

Author

Abdullahi Umar, Feb 28 2013

Keywords

• nonn
• tabl
• easy

Comments

Row sums are A059570.

Examples

			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
  ...
		

Links

• 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

Crossrefs

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

Programs

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

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).