A273715 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k peaks of width 1 (i.e., UHD configurations, where U=(0,1), H(1,0), D=(0,-1)), (n>=2, k>=0).

0, 1, 1, 1, 2, 3, 5, 8, 13, 21, 1, 34, 57, 6, 90, 158, 27, 241, 445, 107, 1, 652, 1269, 396, 10, 1780, 3655, 1404, 66, 4899, 10611, 4838, 356, 1, 13581, 31002, 16344, 1700, 15, 37893, 91048, 54429, 7482, 135, 106340, 268536, 179332, 31070, 940, 1

Offset

2,5

Comments

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

Sum(k*T(n,k),k>=1) = A273716(n).

Links

Alois P. Heinz, Rows n = 2..250, flattened

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

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

Formula

G.f.: G(t,z) satisfies z*G^2 - (1-2*z-z^2-z^3+t*z^3)G + z^2*(t+z-t*z) = 0.

Example

Row 4 is 2,3 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 0,1,1,0,1 peaks of width 1.
Triangle T(n,k) begins:
:    0,    1;
:    1,    1;
:    2,    3;
:    5,    8;
:   13,   21,   1;
:   34,   57,   6;
:   90,  158,  27;
:  241,  445, 107,  1;
:  652, 1269, 396, 10;

Maple

eq := z*G^2-(1-2*z-z^2-z^3+t*z^3)*G+z^2*(t+z-t*z) = 0: G := RootOf(eq, G): 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, h) option remember; expand(
      `if`(n=0, (1-t)*z^h, `if`(t<0, 0, b(n-1, y+1, 1, 0))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0)*z^h)+
      `if`(y<1, 0, b(n-1, y, 0, `if`(t>0, 1, 0)))))
    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 06 2016

Mathematica

b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[ If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + If[y < 1, 0, b[n - 1, y, 0, If[t > 0, 1, 0]]]]] ; 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 *)

Crossrefs

Cf. A082582, A273716.

Sequence in context: A005347 A100582 A193616 * A093093 A345095 A281408

Adjacent sequences: A273712 A273713 A273714 * A273716 A273717 A273718

Keyword

nonn,tabf

Author

Emeric Deutsch, May 28 2016

Status

approved