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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.

%I A221879 #31 Aug 17 2025 03:00:30
%S A221879 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 A221879 42,96,120,88,36,8,0,9,56,154,240,230,136,48,8,1,10,72,232,434,516,
%U A221879 400,200,60,10,0,11,90,333,728,1036,996,650,280,75,10,1
%N 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.
%C A221879 Row sums are A059570.
%H A221879 A. D. Adeshola, V. Maltcev and A. Umar, <a href="http://arxiv.org/abs/1303.7428">Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain</a>, arXiv:1303.7428 [math.CO], 2013.
%H A221879 A. D. Adeshola, A. Umar, <a href="https://combinatorialpress.com/jcmcc/vol106/">Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain</a>, JMCC 106 (2017) 37-49
%F A221879 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.
%F A221879 Sum_{k=1..n} T(n,k) = A059570(n).
%e A221879 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 A221879 Triangle starts:
%e A221879   1,
%e A221879   2, 0,
%e A221879   3, 2, 1,
%e A221879   4, 6, 4, 0,
%e A221879   5, 12, 12, 4, 1,
%e A221879   6, 20, 28, 18, 6, 0,
%e A221879   7, 30, 55, 52, 27, 6, 1,
%e A221879   8, 42, 96, 120, 88, 36, 8, 0,
%e A221879   9, 56, 154, 240, 230, 136, 48, 8, 1,
%e A221879   10, 72, 232, 434, 516, 400, 200, 60, 10, 0,
%e A221879   11, 90, 333, 728, 1036, 996, 650, 280, 75, 10, 1
%e A221879   ...
%p A221879 A221879 := proc(n,k)
%p A221879     option remember ;
%p A221879     if n<1 then
%p A221879         0 ;
%p A221879     elif n=1 then
%p A221879         if k = 1 then
%p A221879             1;
%p A221879         else
%p A221879             0 ;
%p A221879         end if;
%p A221879     else
%p A221879         if n = 2 and k=2 then
%p A221879             0;
%p A221879         else
%p A221879             (n-k+1)*binomial(n-2,k-1)+procname(n-2,k-2) ;
%p A221879         end if;
%p A221879     end if;
%p A221879 end proc:
%p A221879 seq(seq( A221879(n,k),k=1..n),n=1..20) ; # _R. J. Mathar_, Aug 15 2025
%Y A221879 Cf. A059570 (row sums), A221876, A221877, A221878, A221880, A221881, A221882.
%K A221879 nonn,tabl,easy
%O A221879 1,2
%A A221879 _Abdullahi Umar_, Feb 28 2013