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
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] 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
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.
Crossrefs
Programs
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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
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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 *)
Formula
G.f.: t(1-z)(1-2z-z^2-sqrt((1-z)(1-3z-z^2-z^3)))/(2z(1-tz)).
Comments