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

Author

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

Keywords

• nonn
• tabf

Comments

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

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

Examples

			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.
		

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

Crossrefs

Cf. A082582, A273345.

Programs

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 *)

Formula

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