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 A246049 Number T(n,k) of endofunctions on [n] where the smallest cycle length equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 14
 1, 0, 1, 0, 3, 1, 0, 19, 6, 2, 0, 175, 51, 24, 6, 0, 2101, 580, 300, 120, 24, 0, 31031, 8265, 4360, 2160, 720, 120, 0, 543607, 141246, 74130, 41160, 17640, 5040, 720, 0, 11012415, 2810437, 1456224, 861420, 430080, 161280, 40320, 5040 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS T(0,0) = 1 by convention. In general, number of endofunctions on [n] where the smallest cycle length equals k is asymptotic to (exp(-H(k-1)) - exp(-H(k))) * n^n, where H(k) is the harmonic number A001008/A002805, k>=1. - Vaclav Kotesovec, Aug 21 2014 LINKS Alois P. Heinz, Rows n = 0..140, flattened EXAMPLE Triangle T(n,k) begins: 1; 0, 1; 0, 3, 1; 0, 19, 6, 2; 0, 175, 51, 24, 6; 0, 2101, 580, 300, 120, 24; 0, 31031, 8265, 4360, 2160, 720, 120; 0, 543607, 141246, 74130, 41160, 17640, 5040, 720; ... MAPLE with(combinat): b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0, add((i-1)!^j*multinomial(n, n-i*j, i\$j)/j!* b(n-i*j, i+1), j=0..n/i))) end: A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, k), j=0..n): T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), A(n, k) -A(n, k+1)): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, Sum[(i-1)!^j*multinomial[n, {n-i*j, Sequence @@ Table[i, {j}]}]/j!* b[n-i*j, i+1], {j, 0, n/i}]]]; A[n_, k_] := Sum[Binomial[n-1, j-1]*n^(n-j)*b[j, k], {j, 0, n}]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], A[n, k] - A[n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 06 2015, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000007, A045531, A246189, A246190, A246191, A246192, A246193, A246194, A246195, A246196, A246197. T(2n,n) gives A246050. Row sums give A000312. Main diagonal gives A000142(n-1) for n>0. Cf. A241981, A243098. Sequence in context: A241981 A147723 A110518 * A316773 A006837 A158782 Adjacent sequences: A246046 A246047 A246048 * A246050 A246051 A246052 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 11 2014 STATUS approved

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Last modified May 24 13:35 EDT 2024. Contains 372773 sequences. (Running on oeis4.)