A274486 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal segments (n>=2, k>=1). A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0).

1, 2, 3, 2, 4, 8, 1, 5, 20, 10, 6, 40, 45, 6, 7, 70, 140, 56, 2, 8, 112, 350, 280, 44, 9, 168, 756, 1008, 366, 20, 10, 240, 1470, 2940, 1920, 320, 5, 11, 330, 2640, 7392, 7590, 2552, 190, 12, 440, 4455, 16632, 24684, 13904, 2445, 70

Offset

2,2

Comments

Sum of entries in row n = A082582(n).

Sum(k*T(n,k), k>=0) = A273345(n+1).

Links

Alois P. Heinz, Rows n = 2..200, flattened

M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Formula

G.f.: G = (1-2z+z^2-2tz^2-sqrt((1-z)((1-z)^3-4tz^2*(1-z+tz))))/(2tz).

Example

Row 4 is 3,2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1],[1,2],[2,1],[2,2],[3] and, clearly, they have 1,2,2,1,1 horizontal segments.
Triangle starts
1;
2;
3,2;
4,8,1;
5,20,10;
6,40,45,6.

Maple

G := ((1-2*z+z^2-2*t*z^2-sqrt((1-z)*((1-z)^3-4*t*z^2*(1-z+t*z))))*(1/2))/(t*z): Gser := simplify(series(G, z = 0, 23)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1))+
      `if`(y<1, 0, b(n-1, y, 0)*`if`(t=0, 1, z))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..20);  # Alois P. Heinz, Jun 27 2016

Mathematica

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, 1 - t, If[t < 0, 0, b[n - 1, y + 1, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]] + If[y < 1, 0, b[n - 1, y, 0]*If[t == 0, 1, z]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Dec 02 2016, after Alois P. Heinz *)

Crossrefs

Cf. A082582, A273345.

Sequence in context: A338902 A116217 A333907 * A227961 A108838 A318176

Adjacent sequences: A274483 A274484 A274485 * A274487 A274488 A274489

Keyword

nonn,tabf

Author

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

Status

approved