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 A221876 T(n,k) is the number of order-preserving full contraction mappings (of an n-chain) with exactly k fixed points. 9
 1, 2, 1, 5, 2, 1, 12, 5, 2, 1, 28, 12, 5, 2, 1, 64, 28, 12, 5, 2, 1, 144, 64, 28, 12, 5, 2, 1, 320, 144, 64, 28, 12, 5, 2, 1, 704, 320, 144, 64, 28, 12, 5, 2, 1, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sum is A001792(n-1). The matrix inverse starts 1; -2,1; -1,-2,1; 0,-1,-2,1; 1,0,-1,-2,1; 2,1,0,-1,-2,1; 3,2,1,0,-1,-2,1; 4,3,2,1,0,-1,-2,1; 5,4,3,2,1,0,-1,-2,1; 6,5,4,3,2,1,0,-1,-2,1; 7,6,5,4,3,2,1,0,-1,-2,1; - R. J. Mathar, Apr 12 2013 ... T(n,k) is also the total number of occurrences of parts k in all compositions (ordered partitions) of n, see example. The equivalent sequence for partitions is A066633. Omar E. Pol, Aug 26 2013 REFERENCES A. D. Adeshola, V. Maltcev and A. Umar, Combinatorial results for certain semigroups of order-preserving full contraction mappings of a finite chain, (submitted 2013). LINKS Table of n, a(n) for n=1..66. 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,n) = 1, T(n,k) = (n-k+3)*2^(n-k-2) for n>=2 and n > k > 0. T(2*n+1,n+1) = T(n+1,1) = A045623(n) for n>=0. T(n,k) = A045623(n-k), n>=1, 1<=k<=n. - Omar E. Pol, Sep 01 2013 EXAMPLE T(5,3) = 5 because there are exactly 5 order-preserving full contraction mappings (of a 5-chain) with exactly 3 fixed points, namely: (12333), (12334), (22344), (23345), (33345). Triangle begins: 1, 2, 1, 5, 2, 1, 12, 5, 2, 1, 28, 12, 5, 2, 1, 64, 28, 12, 5, 2, 1, 144, 64, 28, 12, 5, 2, 1, 320, 144, 64, 28, 12, 5, 2, 1, 704, 320, 144, 64, 28, 12, 5, 2, 1, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1, 3328, 1536, 704, 320, 144, 64, 28, 12, 5, 2, 1, ... Note that column k is column 1 shifted down by k positions. Row 4 is [12, 5, 2, 1]: in the compositions of 4 [ 1] [ 1 1 1 1 ] [ 2] [ 1 1 2 ] [ 3] [ 1 2 1 ] [ 4] [ 1 3 ] [ 5] [ 2 1 1 ] [ 6] [ 2 2 ] [ 7] [ 3 1 ] [ 8] [ 4 ] there are 12 parts=1, 5 parts=2, 2 part=3, and 1 part=4. - Joerg Arndt, Sep 01 2013 MATHEMATICA T[n_, n_] = 1; T[n_, k_] := (n - k + 3)*2^(n - k - 2); Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 21 2018 *) CROSSREFS Cf. A001792, A221877, A221878, A221879, A221880, A221881, A221882. Sequence in context: A361681 A105084 A126125 * A128514 A323953 A126075 Adjacent sequences: A221873 A221874 A221875 * A221877 A221878 A221879 KEYWORD nonn,easy,tabl AUTHOR Abdullahi Umar, Feb 28 2013 STATUS approved

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Last modified June 20 13:44 EDT 2024. Contains 373527 sequences. (Running on oeis4.)