A097971 -id:A097971 - cp's OEIS frontend

A120928 Number of "ups" and "downs" in the permutations of [n] if either a previous counted "up" ("down") or a "void" precedes an "up" ("down") which then will be counted also.

2, 8, 44, 280, 2040, 16800, 154560, 1572480, 17539200, 212889600, 2794176000, 39437798400, 595718323200, 9589612032000, 163895187456000, 2964061900800000, 56554301067264000, 1135354270482432000, 23923536413736960000, 527939735774330880000

Offset: 2

Thomas Wieder, Jul 16 2006

An "up" ("down") is a neighboring pair of elements e_i, e_j of [n] with e_i < e_j (e_i > e_j). A "void" is a missing preceding pair, i.e., the start of [n]. We discuss two examples for [n=4]. In the permutation [3, 1, 2, 4] "void" precedes the pair 3,1 and consequently a "down" is counted. No "up" which has been counted precedes the "ups" 1,2 and 2,4 so they are not counted. In [3, 4, 1, 2] the "up" 3,4 is counted and so is the next "up" 1,2 but the down 4,1 has no preceding "down" registered and is therefore not counted.

			[1, 2, 3, 4], "ups"=3, "downs"=0;
[1, 2, 4, 3], "ups"=2, "downs"=0;
[1, 3, 2, 4], "ups"=2, "downs"=0;
[1, 3, 4, 2], "ups"=2, "downs"=0;
[1, 4, 2, 3], "ups"=2, "downs"=0;
[1, 4, 3, 2], "ups"=1, "downs"=0;
[2, 1, 3, 4], "ups"=0, "downs"=1;
[2, 1, 4, 3], "ups"=0, "downs"=2;
[2, 3, 1, 4], "ups"=2, "downs"=0;
[2, 3, 4, 1], "ups"=2, "downs"=0;
[2, 4, 1, 3], "ups"=2, "downs"=0;
[2, 4, 3, 1], "ups"=1, "downs"=0;
[3, 1, 2, 4], "ups"=0, "downs"=1;
[3, 1, 4, 2], "ups"=0, "downs"=2;
[3, 2, 1, 4], "ups"=0, "downs"=2;
[3, 2, 4, 1], "ups"=0, "downs"=2;
[3, 4, 1, 2], "ups"=2, "downs"=0;
[3, 4, 2, 1], "ups"=1, "downs"=0;
[4, 1, 2, 3], "ups"=0, "downs"=1;
[4, 1, 3, 2], "ups"=0, "downs"=2;
[4, 2, 1, 3], "ups"=0, "downs"=2;
[4, 2, 3, 1], "ups"=0, "downs"=2;
[4, 3, 1, 2], "ups"=0, "downs"=2;
[4, 3, 2, 1], "ups"=0, "downs"=3.

Alois P. Heinz, Table of n, a(n) for n = 2..400

Cf. A028399, A052582, A062119, A097971.

a:= n-> ceil(n!*(3*n-1)/6):
seq(a(n), n=2..30); # Alois P. Heinz, Apr 21 2012

E.g.f.: -(6+6*x^2-4*x^3+x^4)/(-3+12*x-18*x^2+12*x^3-3*x^4). - Thomas Wieder, May 02 2009

a(2) = 2, a(n) = n! * (3*n - 1) / 6 for n > 2. - Jon E. Schoenfield, Apr 18 2010

4 more terms from R. J. Mathar, Aug 25 2008

More terms from Alois P. Heinz, Apr 21 2012

A186371 Number of up-down runs in all permutations of {1,2,...,n}.

Original entry on oeis.org

0, 1, 3, 13, 68, 420, 3000, 24360, 221760, 2237760, 24796800, 299376000, 3911846400, 55005350400, 828193766400, 13294689408000, 226663557120000, 4090405423104000, 77895546753024000, 1561112121913344000, 32844177110384640000, 723788347432550400000

Offset: 0

Emeric Deutsch and Ira M. Gessel, Mar 01 2011

The up-down runs of a permutation p are the alternating runs of the permutation p endowed with a 0 in the front. For example, 75814632 has 6 up-down runs: 07, 75, 58, 81, 146, and 632.

			a(3)=13 because the permutations 123, 132, 213, 231, 312, and 321 have a total of 1 + 2 + 3 + 2 + 3 + 2 = 13 up-down runs.

Cf. A186370, A097971.

[0,1] cat [Factorial(n)*(4*n+1)/6: n in [2..30]]; // Vincenzo Librandi, Sep 11 2015

0, 1, seq((1/6)*factorial(n)*(4*n+1), n = 2 .. 20);

Join[{0, 1}, Table[n! (4 n + 1)/6, {n, 2, 20}]] (* Vincenzo Librandi, Sep 11 2015 *)

a(n) = Sum_{k=1..n} k*A186370(n,k).
a(n) = n!*(4n+1)/6 for n>=2.
E.g.f.: g(z) = z(6-3z+z^2)/[6(1-z)^2].
D-finite with recurrence 4*a(n) +(-4*n-7)*a(n-1) +3*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 22 2022

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A120928 Number of "ups" and "downs" in the permutations of [n] if either a previous counted "up" ("down") or a "void" precedes an "up" ("down") which then will be counted also.

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