A273717 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k L-shaped corners (n>=2, k>=0).

1, 2, 4, 1, 8, 5, 16, 18, 1, 32, 56, 9, 64, 160, 50, 1, 128, 432, 220, 14, 256, 1120, 840, 110, 1, 512, 2816, 2912, 645, 20, 1024, 6912, 9408, 3150, 210, 1, 2048, 16640, 28800, 13552, 1575, 27, 4096, 39424, 84480, 53088, 9534, 364, 1, 8192, 92160, 239360, 193440, 49644, 3388, 35, 16384, 212992, 658944, 665280, 231000, 24822, 588

Offset: 2

Author

Emeric Deutsch, May 29 2016

Keywords

• nonn
• tabf

Comments

Each L-shaped corner can be viewed as a descent (as defined in Sec. 5.1 of the Blecher et al. reference). - Emeric Deutsch, Jul 02 2016

Examples

			Row 4 is 4,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] of which only [2,1] yields a |_ - shaped corner.
Triangle starts:
  1;
  2;
  4,1;
  8,5;
  16,18,1.
		

References

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

Links

• Alois P. Heinz, Rows n = 2..200, flattened
• Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.
• M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. Appl. Math., 31, 2003, 86-112.
• Emeric Deutsch and S. Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088 [math.CO], 2016.

Crossrefs

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

Cf. A001793, A273718.

Programs

Maple

eq := t*z*G^2-(1-2*z-t*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): 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 .. 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, z, 1))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..18);  # Alois P. Heinz, Jun 06 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, z, 1]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 18}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

Formula

G.f.: G = G(t,z) satisfies t*z*G^2 - (1 - 2*z - t*z^2)*G + z^2 = 0.

Sum_{k>=1} k*T(n,k) = A273718(n).

T(n,0) = 2^(n-2).

T(n,1) = n*(n-3)*2^(n-6) = A001793(n-3) for n>=4.