A273344 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k levels. A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step.

1, 1, 1, 3, 2, 6, 7, 14, 19, 2, 33, 53, 11, 79, 148, 47, 1, 194, 409, 181, 10, 482, 1137, 639, 69, 1214, 3159, 2166, 360, 6, 3090, 8793, 7110, 1646, 66, 7936, 24515, 22831, 6868, 490, 2, 20544, 68443, 72145, 26893, 2918, 44, 53545, 191367, 225138, 100598, 15085, 486, 140399, 535762, 695798, 363360, 70847, 3825

Offset: 2

Author

Emeric Deutsch, May 21 2016

Keywords

• nonn
• tabf

Comments

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

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] having 1, 0, 0, 1, 0 levels, respectively.
Triangle starts
1;
1,1;
3,2;
6,7;
14,19,2.
		

Links

• Alois P. Heinz, Rows n = 2..250, flattened
• A. Blecher, C. Brennan, and A. Knopfmacher, Levels in bargraphs, Ars Math. Contemp., 9, 2015, 297-310.
• A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A025243, A082582, A273345.

Programs

Maple

G := (1-2*z-z^2+2*z^3-2*t*z^3-sqrt((1-z)*(1-3*z-z^2+3*z^3-4*t*z^3+4*z^4 -4*t*z^4-4*z^5+8*t*z^5-4*t^2*z^5)))/(2*z*(1-z+t*z)): Gser := simplify(series(G, z = 0, 25)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t, w) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+ `if`(y<1, 0,
      `if`(w=1, z, 1)*b(n-1, y, 0, min(w+1, 2)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..20);  # Alois P. Heinz, Jun 04 2016

Mathematica

b[n_, y_, t_, w_] := b[n, y, t, w] = Expand[If[n == 0, (1 - t), If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, If[w == 1, z, 1]*b[n - 1, y, 0, Min[w + 1, 2]]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)

Formula

T(n,0) = A025243(n+1).

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

G.f.: G(t,z) = (1-2z-z^2+2z^3-2tz^3-sqrt((1-z)(1-3z-z^2+3z^3-4tz^3+4z^4-4tz^4-4z^5+8tz^5-4t^2z^5)))/(2z(1-z+tz)); z marks semiperimeter, t marks levels. See eq. (2.4) in the Blecher et al. Ars. Math. Contemp. reference (set x = z, y = z, w = t).