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.
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, 24, 2, 8, 98, 252, 350, 280, 126, 28, 2, 9, 128, 392, 672, 700, 448, 168, 32, 2, 10, 162, 576, 1176, 1512, 1260, 672, 216, 36, 2
Offset: 1
Examples
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). From _Paolo Xausa_, Aug 18 2025: (Start) Triangle begins: 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, 24, 2; 8, 98, 252, 350, 280, 126, 28, 2; 9, 128, 392, 672, 700, 448, 168, 32, 2; 10, 162, 576, 1176, 1512, 1260, 672, 216, 36, 2; ... (End)
Links
- Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of triangle, flattened).
- 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
Programs
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Mathematica
A221877[n_, k_] := If[k == 1, n, 2*(n-k+1)*Binomial[n-1, k-1]]; Table[A221877[n, k], {n, 15}, {k, n}] (* Paolo Xausa, Aug 18 2025 *)
Formula
T(n,1) = n and T(n,k) = 2(n-k+1)*C(n-1,k-1) if k > 1.
Extensions
Name edited by Paolo Xausa, Aug 18 2025
Comments