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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.
5

%I #24 Mar 30 2013 14:31:51

%S 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,

%T 42,96,120,88,36,8,0,9,56,154,240,230,136,48,8,1,10,72,232,434,516,

%U 400,200,60,10,0,11,90,333,728,1036,996,650,280,75,10,1

%N 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.

%C Row sums are A059570.

%D A. D. Adeshola, V. Maltcev and A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, (submitted).

%F 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.

%e 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).

%e Triangle starts

%e 1,

%e 2, 0,

%e 3, 2, 1,

%e 4, 6, 4, 0,

%e 5, 12, 12, 4, 1,

%e 6, 20, 28, 18, 6, 0,

%e 7, 30, 55, 52, 27, 6, 1,

%e 8, 42, 96, 120, 88, 36, 8, 0,

%e 9, 56, 154, 240, 230, 136, 48, 8, 1,

%e 10, 72, 232, 434, 516, 400, 200, 60, 10, 0,

%e 11, 90, 333, 728, 1036, 996, 650, 280, 75, 10, 1

%e ...

%Y Cf. A059570, A221876, A221877, A221878, A221880, A221881, A221882.

%K nonn,tabl

%O 1,2

%A _Abdullahi Umar_, Feb 28 2013