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A183154 T(n,k) is the number of order-preserving partial isometries (of an n-chain) of fixed k (fix of alpha is the number of fixed points of alpha) 3
1, 1, 1, 3, 2, 1, 9, 3, 3, 1, 23, 4, 6, 4, 1, 53, 5, 10, 10, 5, 1, 115, 6, 15, 20, 15, 6, 1, 241, 7, 21, 35, 35, 21, 7, 1, 495, 8, 28, 56, 70, 56, 28, 8, 1, 1005, 9, 36, 84, 126, 126, 84, 36, 9, 1, 2027, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
R. Kehinde and A. Umar, On the semigroup of partial isometries of a finite chain, arXiv:1101.2558 [math.GR], 2011.
FORMULA
T(n,0) = A183155(n) and T(n,k) = binomial(n,k) if k > 0.
EXAMPLE
T (4,2) = 6 because there are exactly 6 order-preserving partial isometries (on a 4-chain) of fix 2, namely: (1,2)-->(1,2); (2,3)-->(2,3); (3,4)-->(3,4); (1,3)-->(1,3); (2,4)-->(2,4); (1,4)-->(1,4) - the mappings are coordinate-wise.
Triangle starts as:
1;
1, 1;
3, 2, 1;
9, 3, 3, 1;
23, 4, 6, 4, 1;
53, 5, 10, 10, 5, 1;
115, 6, 15, 20, 15, 6, 1;
MAPLE
A183155 := proc(n) 2^(n+1)-2*n-1 ; end proc:
A183154 := proc(n, k) if k =0 then A183155(n); else binomial(n, k) ; end if; end proc: # R. J. Mathar, Jan 06 2011
MATHEMATICA
T[n_, k_] := If[k == 0, 2^(n + 1) - 2n - 1, Binomial[n, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 17 2018 *)
PROG
(PARI) A183155(n)=2^(n+1) - (2*n+1);
T(n, k)=if(k==0, A183155(n), binomial(n, k));
for(n=0, 17, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Joerg Arndt, Dec 30 2010
CROSSREFS
Sequence in context: A086963 A079749 A156647 * A193791 A160760 A152860
KEYWORD
nonn,easy,tabl
AUTHOR
Abdullahi Umar, Dec 28 2010
STATUS
approved

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Last modified June 18 01:52 EDT 2024. Contains 373468 sequences. (Running on oeis4.)