A273897 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having abscissa of first descent k (n>=2, 1<=k<=n-1).

1, 1, 1, 2, 2, 1, 4, 5, 3, 1, 9, 12, 9, 4, 1, 22, 30, 25, 14, 5, 1, 57, 78, 69, 44, 20, 6, 1, 154, 210, 192, 133, 70, 27, 7, 1, 429, 582, 542, 396, 230, 104, 35, 8, 1, 1223, 1651, 1554, 1176, 731, 369, 147, 44, 9, 1, 3550, 4772, 4521, 3504, 2285, 1248, 560, 200, 54, 10, 1

Offset: 2

Author

Emeric Deutsch, Jun 06 2016

Keywords

• nonn
• tabl

Comments

Number of entries in row n is n-1.

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

T(n,1) = A105633(n-3) (n>=3).

Sum(k*T(n,k), k>=1) = A273898(n).

Examples

			Row 4 is 2,2,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding pictures give the values 3,2,1,2,1 for the abscissae of the first descents.
Triangle starts
1;
1,1;
2,2,1;
4,5,3,1;
9,12,9,4,1;
22,30,25,14,5,1.
		

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

Crossrefs

Cf. A082582, A105633, A273898.

Programs

Maple

G := (1/2)*t*z*(1-2*t*z-z^2-sqrt(1-4*z+2*z^2+z^4))/(1-t-z+t^2*z+t*z^2): Gser := simplify(series(G, z = 0, 20)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 18 do seq(coeff(P[n], t, j), j = 1 .. n-1) end do; # yields sequence in triangular form

Mathematica

nmax = 13; G = (1/2) t z (1 - 2t z - z^2 - Sqrt[1 - 4z + 2z^2 + z^4])/(1 - t - z + t^2 z + t z^2); Gser = G + O[z]^nmax;
Do[P[n] = Expand[Coefficient[Gser, z, n]], {n, 2, nmax}];
Table[CoefficientList[P[n]/t, t], {n, 2, nmax}] // Flatten (* Jean-François Alcover, Jul 24 2018, from Maple *)

Formula

G.f.: G(t,z)=(1/2)tz(1-2tz-z^2-sqrt(1-4z+2z^2+z^4))/(1-t-z+t^2z+tz^4), where z marks semiperimeter and t marks the abscissa of the first descent.