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URL: https://oeis.org/A221877

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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.
7
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
1,2
COMMENTS
Row sums are A221882.
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.
FORMULA
T(n,1) = n and T(n,k) = 2(n-k+1)*C(n-1,k-1) if k > 1.
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Abdullahi Umar, Feb 28 2013
EXTENSIONS
Name edited by Paolo Xausa, Aug 18 2025
STATUS
approved