A273349 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k level steps (n>=2,k>=0). A level step in a bargraph is any pair of adjacent horizontal steps at the same height.

1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 14, 12, 7, 1, 1, 33, 34, 19, 9, 1, 1, 79, 95, 61, 27, 11, 1, 1, 194, 261, 193, 95, 36, 13, 1, 1, 482, 728, 585, 333, 136, 46, 15, 1, 1, 1214, 2022, 1797, 1091, 521, 184, 57, 17, 1, 1, 3090, 5634, 5439, 3629, 1821, 763, 239, 69, 19, 1, 1

Offset: 2

Author

Emeric Deutsch, Jun 03 2016

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n-1.

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

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

Sum(k*T(n,k),k>=0) = A271943(n-1). This implies that the number of level steps in all bargraphs of semiperimeter n is equal to the sum of the widths of all bargraphs of semiperimeter n-1.

Examples

			Row 4 is 3,1,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] which, clearly, have 2,0,0,1,0 level steps.
Triangle starts
1;
1,1;
3,1,1;
6,5,1,1;
14,12,7,1,1
		

References

• A. Blecher, C. Brennan, and A. Knopfmacher, Combinatorial parameters in bargraphs (preprint).

Links

• Alois P. Heinz, Rows n = 2..150, flattened
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Crossrefs

Cf. A025243, A082582, A271943.

Programs

Maple

G:=((1-t*z-z-2*z^2+t*z^2-sqrt((1-t*z-z-2*z^2+t*z^2)^2-4*z^3))*(1/2))/z: Gser:=simplify(series(G,z=0,21)): for n from 2 to 18 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 0 .. n-2) 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, 1)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..18);  # 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, 1]]]]];
T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][ b[n, 0, 0, 0]];
Table[T[n], {n, 2, 18}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)

Formula

G.f.: G(t,z) = (1-tz-z-2z^2+tz^2-sqrt((1-z)(1-z-2tz-4z^2+t^2z^2+2tz^2-4z^3-t^2z^3+4tz^3)))/(2z) (z marks semiperimeter, t marks level steps; obtained from the expression for F in the Blecher et al. reference (Section 7.1) by setting x=z, y=z, w=t).