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 A240796 Total number of occurrences of the pattern 1<2 in all preferential arrangements (or ordered partitions) of n elements. 5
 0, 1, 15, 186, 2330, 31065, 447405, 6979588, 117745668, 2141106795, 41810587775, 873474855726, 19451904450654, 460209050303821, 11531197020389025, 305122289460210120, 8503747639606509128, 249020038061419770783, 7645072502094118876755, 245564189847880300238290 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS There are A000670(n) preferential arrangements of n elements - see A000670, A240763. The number that avoid the pattern 1<2 is 2^(n-1). The total number of occurrences of the pattern 1<2 in all permutations on n elements is (n-1)*(n-1)! (cf. A010027, A001563). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..420 FORMULA a(n) ~ n! * n^2 / (8 * (log(2))^(n+1)). - Vaclav Kotesovec, May 03 2015 EXAMPLE The 13 preferential arrangements on 3 points and the number of times the pattern 1<2 occurs are: 1<2<3, 3 1<3<2, 2 2<1<3, 2 2<3<1, 1 3<1<2, 1 3<2<1, 0 1=2<3, 2 1=3<2, 1 2=3<1, 0 1<2=3, 2 2<1=3, 1 3<1=2, 0 1=2=3, 0, for a total of a(3) = 15. MAPLE b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+ [0, p[1]*j*t/2])(b(n-j, t+j))*binomial(n, j), j=1..n)) end: a:= n-> b(n, 0)[2]: seq(a(n), n=1..25); # Alois P. Heinz, Dec 08 2014 MATHEMATICA b[n_, t_] := b[n, t] = If[n == 0, {1, 0}, Sum[Function[{p}, p + {0, p[[1]]*j*t/2}][b[n - j, t + j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 08 2015, after Alois P. Heinz *) CROSSREFS Cf. A000670, A240763, A240796-A240800, A010027, A001563. Sequence in context: A079516 A206521 A366662 * A206514 A016207 A286347 Adjacent sequences: A240793 A240794 A240795 * A240797 A240798 A240799 KEYWORD nonn AUTHOR N. J. A. Sloane, Apr 13 2014 EXTENSIONS a(8)-a(20) from Alois P. Heinz, Dec 08 2014 STATUS approved

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Last modified September 11 10:08 EDT 2024. Contains 375827 sequences. (Running on oeis4.)