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 A347999 Triangular array read by rows: T(n,k) is the number of endofunctions f:{1,2,...,n}-> {1,2,...,n} whose smallest connected component has exactly k nodes; n >= 0, 0 <= k <= n. 6
 1, 0, 1, 0, 1, 3, 0, 10, 0, 17, 0, 87, 27, 0, 142, 0, 1046, 510, 0, 0, 1569, 0, 15395, 6795, 2890, 0, 0, 21576, 0, 269060, 114912, 84490, 0, 0, 0, 355081, 0, 5440463, 2332029, 1493688, 705740, 0, 0, 0, 6805296, 0, 124902874, 53389746, 32186168, 28072548, 0, 0, 0, 0, 148869153 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Here component means weakly connected component in the functional digraph of f. If the mapping has no component, then the smallest component is defined to have size 0. For the statistic "length of the largest component", see A209324. REFERENCES R. Sedgewick and P. Flajolet, Analysis of Algorithms, Addison Wesley, 1996, Chapter 8. LINKS Alois P. Heinz, Rows n = 0..140, flattened Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021. D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432. FORMULA T(n,n) = A001865(n) for n >= 1. Sum_{k=1..n} k * T(n,k) = A350157(n). - Alois P. Heinz, Dec 17 2021 EXAMPLE Triangle begins:   1;   0,       1;   0,       1,       3;   0,      10,       0,      17;   0,      87,      27,       0,    142;   0,    1046,     510,       0,      0, 1569;   0,   15395,    6795,    2890,      0,    0, 21576;   0,  269060,  114912,   84490,      0,    0,     0, 355081;   0, 5440463, 2332029, 1493688, 705740,    0,     0,      0, 6805296;   ... MAPLE g:= proc(n) option remember; add(n^(n-j)*(n-1)!/(n-j)!, j=1..n) end: b:= proc(n, m) option remember; `if`(n=0, x^m, add(g(i)*       b(n-i, min(m, i))*binomial(n-1, i-1), i=1..n))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n\$2)): seq(T(n), n=0..12);  # Alois P. Heinz, Dec 16 2021 MATHEMATICA g[n_] := g[n] = Sum[n^(n - j)*(n - 1)!/(n - j)!, {j, 1, n}]; b[n_, m_] := b[n, m] = If[n == 0, x^m, Sum[g[i]*b[n - i, Min[m, i]]* Binomial[n - 1, i - 1], {i, 1, n}]]; T[n_] := With[{p = b[n, n]},  Table[Coefficient[p, x, i], {i, 0, n}]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *) CROSSREFS Columns k=0-1 give: A000007, A350134. Row sums give A000312. Right border gives A001865. T(2n,n) gives A350135. Cf. A209324, A350157. Sequence in context: A336710 A294106 A094472 * A028850 A138364 A095364 Adjacent sequences:  A347996 A347997 A347998 * A348000 A348001 A348002 KEYWORD nonn,tabl AUTHOR Steven Finch, Sep 23 2021 EXTENSIONS Edited by Alois P. Heinz, Dec 15 2021 STATUS approved

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Last modified January 22 23:50 EST 2022. Contains 350504 sequences. (Running on oeis4.)