A274488 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having least column-height k (n>=2, k>=1).

1, 1, 1, 3, 1, 1, 8, 3, 1, 1, 22, 8, 3, 1, 1, 62, 22, 8, 3, 1, 1, 178, 62, 22, 8, 3, 1, 1, 519, 178, 62, 22, 8, 3, 1, 1, 1533, 519, 178, 62, 22, 8, 3, 1, 1, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 1, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 1, 41937, 13800, 4578, 1533, 519, 178, 62, 22, 8, 3, 1, 1

Offset

2,4

Comments

T(n,k) = number of bargraphs of semiperimeter n for which the width of the leftmost horizontal segment is k. A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0). Example: T(4,1)=3 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the widths of their leftmost horizontal segments are 3, 1, 1, 2, 1.

Number of entries in row n is n-1.

Links

G. C. Greubel, Rows n=2..102 of triangle, 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:1609.00088 [math.CO], 2016/2018.

Formula

G.f.: t(1-z)(1-2z-z^2-sqrt((1-z)(1-3z-z^2-z^3)))/(2z(1-tz)).

Example

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] and, clearly, their least column-heights are 1,1,1,2,3.
Triangle starts
1;
1,1;
3,1,1;
8,3,1,1;
22,8,3,1,1

Maple

G:=(1/2)*t*(1-z)*(1-2*z-z^2-sqrt((1-z)*(1-3*z-z^2-z^3)))/(z*(1-t*z)): Gser:= simplify(series(G, z=0, 28)):for n from 2 to 20 do P[n]:= sort(coeff(Gser, z, n)) end do: for n from 2 to 15 do seq(coeff(P[n], t, k), k=1..n-1) end do; # yields sequence in triangular form

Mathematica

gf = t(1-z)((1 - 2z - z^2 - Sqrt[(1-z)(1 - 3z - z^2 - z^3)])/(2z(1 - t z)));
Rest[CoefficientList[#, t]]& /@ Drop[CoefficientList[gf + O[z]^14, z], 2] // Flatten (* Jean-François Alcover, Nov 16 2018 *)

Crossrefs

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

T(n,1) = A188464(n-3)(n>=3).

Sum(k*T(n,k),k>=1)= A008909(n).

Cf. A273350, A274490.

Sequence in context: A198618 A121461 A273719 * A351889 A203717 A143953

Adjacent sequences: A274485 A274486 A274487 * A274489 A274490 A274491

Keyword

nonn,tabl

Author

Emeric Deutsch, Jul 01 2016

Status

approved