A242153 - OEIS

A242153

Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I #26 Jan 14 2015 12:44:06

%S 1,1,0,1,1,0,2,2,1,0,5,6,3,1,0,16,20,12,4,1,0,61,80,50,20,5,1,0,271,

%T 366,240,100,30,6,1,0,1372,1897,1281,560,175,42,7,1,0,7795,10976,7588,

%U 3416,1120,280,56,8,1,0,49093,70155,49392,22764,7686,2016,420,72,9,1,0

%N Number T(n,k) of ascent sequences of length n with exactly k flat steps; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%C In general, column k is asymptotic to Pi^(2*k-5/2) / (k! * 6^(k-2) * sqrt(3) * exp(Pi^2/12)) * (6/Pi^2)^n * n! * sqrt(n). - _Vaclav Kotesovec_, Aug 27 2014

%H Joerg Arndt and Alois P. Heinz, <a href="/A242153/b242153.txt">Rows n = 0..140, flattened</a>

%e Triangle T(n,k) begins:

%e 00: 1;

%e 01: 1, 0;

%e 02: 1, 1, 0;

%e 03: 2, 2, 1, 0;

%e 04: 5, 6, 3, 1, 0;

%e 05: 16, 20, 12, 4, 1, 0;

%e 06: 61, 80, 50, 20, 5, 1, 0;

%e 07: 271, 366, 240, 100, 30, 6, 1, 0;

%e 08: 1372, 1897, 1281, 560, 175, 42, 7, 1, 0;

%e 09: 7795, 10976, 7588, 3416, 1120, 280, 56, 8, 1, 0;

%e 10: 49093, 70155, 49392, 22764, 7686, 2016, 420, 72, 9, 1, 0;

%e ...

%e The 15 ascent sequences of length 4 (dots denote zeros) with their number of flat steps are:

%e 01: [ . . . . ] 3

%e 02: [ . . . 1 ] 2

%e 03: [ . . 1 . ] 1

%e 04: [ . . 1 1 ] 2

%e 05: [ . . 1 2 ] 1

%e 06: [ . 1 . . ] 1

%e 07: [ . 1 . 1 ] 0

%e 08: [ . 1 . 2 ] 0

%e 09: [ . 1 1 . ] 1

%e 10: [ . 1 1 1 ] 2

%e 11: [ . 1 1 2 ] 1

%e 12: [ . 1 2 . ] 0

%e 13: [ . 1 2 1 ] 0

%e 14: [ . 1 2 2 ] 1

%e 15: [ . 1 2 3 ] 0

%e There are 5 sequences without flat steps, 6 with one flat step, etc., giving row [5, 6, 3, 1, 0] for n=4.

%p b:= proc(n, i, t) option remember; `if`(n=0, 1, expand(add(

%p `if`(j=i, x, 1) *b(n-1, j, t+`if`(j>i, 1, 0)), j=0..t+1)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, -1$2)):

%p seq(T(n), n=0..12);

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Expand[Sum[If[j == i, x, 1]*b[n-1, j, t + If[j>i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][ b[n, -1, -1]]; Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 06 2015, after _Alois P. Heinz_ *)

%Y Columns k=0-10 give: A138265, A242154, A242155, A242156, A242157, A242158, A242159, A242160, A242161, A242162, A242163.

%Y Row sums give A022493.

%Y T(2n,n) gives A242164.

%Y Main diagonal and lower diagonals give: A000007, A000012, A000027(n+1), A002378(n+1), A134481(n+1), A130810(n+4).

%Y Cf. A137251 (the same for ascents), A238858 (the same for descents).

%K nonn,tabl

%O 0,7

%A _Joerg Arndt_ and _Alois P. Heinz_, May 05 2014