cp's OEIS Frontend

A221877 Triangle read by rows: T(n,k) = number of order-preserving or order-reversing full contraction mappings (of an n-chain) with height exactly k.

%I A221877 #27 Aug 18 2025 06:28:24
%S A221877 1,2,2,3,8,2,4,18,12,2,5,32,36,16,2,6,50,80,60,20,2,7,72,150,160,90,
%T A221877 24,2,8,98,252,350,280,126,28,2,9,128,392,672,700,448,168,32,2,10,162,
%U A221877 576,1176,1512,1260,672,216,36,2
%N A221877 Triangle read by rows: T(n,k) = number of order-preserving or order-reversing full contraction mappings (of an n-chain) with height exactly k.
%C A221877 Row sums are A221882.
%H A221877 Paolo Xausa, <a href="/A221877/b221877.txt">Table of n, a(n) for n = 1..11325</a> (rows 1..150 of triangle, flattened).
%H A221877 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 A221877 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 A221877 T(n,1) = n and T(n,k) = 2(n-k+1)*C(n-1,k-1) if k > 1.
%e A221877 T(3,2) = 8 because there are exactly 8 order-preserving full contraction mappings (of a 3-chain) with exactly height 2, namely: (112), (122), (211), (221), (223), (233), (322), (332).
%e A221877 From _Paolo Xausa_, Aug 18 2025: (Start)
%e A221877 Triangle begins:
%e A221877    1;
%e A221877    2,   2;
%e A221877    3,   8,   2;
%e A221877    4,  18,  12,    2;
%e A221877    5,  32,  36,   16,    2;
%e A221877    6,  50,  80,   60,   20,    2;
%e A221877    7,  72, 150,  160,   90,   24,   2;
%e A221877    8,  98, 252,  350,  280,  126,  28,   2;
%e A221877    9, 128, 392,  672,  700,  448, 168,  32,  2;
%e A221877   10, 162, 576, 1176, 1512, 1260, 672, 216, 36, 2;
%e A221877   ... (End)
%t A221877 A221877[n_, k_] := If[k == 1, n, 2*(n-k+1)*Binomial[n-1, k-1]];
%t A221877 Table[A221877[n, k], {n, 15}, {k, n}] (* _Paolo Xausa_, Aug 18 2025 *)
%Y A221877 Cf. A221876, A221878, A221879, A221880, A221881, A221882.
%K A221877 nonn,tabl
%O A221877 1,2
%A A221877 _Abdullahi Umar_, Feb 28 2013
%E A221877 Name edited by _Paolo Xausa_, Aug 18 2025