A273721 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k valleys of width 1 (i.e., DHU configurations, where U=(0,1), H=(1,0), D=(0,-1)), (n>=2, k>=0).
1, 2, 5, 13, 34, 1, 90, 7, 241, 34, 652, 141, 1, 1780, 536, 11, 4899, 1929, 77, 13581, 6690, 433, 1, 37893, 22601, 2132, 16, 106340, 74898, 9598, 151, 299978, 244632, 40517, 1091, 1, 850187, 790144, 162988, 6670, 22, 2419788, 2529873, 631655, 36330, 268
Offset: 2
Examples
Row 4 is 5 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 drawings show that they have no 1-width valleys. T(6,1) = 1 because there is only one bargraph of semiperimeter 6 having a 1-width valley (it corresponds to the composition [2,1,2]). Triangle starts 1; 2; 5; 13; 34, 1;
Links
- Alois P. Heinz, Rows n = 2..250, 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
Programs
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Maple
eq := z*(t+z-t*z)*G^2-(1-2*z-z^2-z^3+t*z^3)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 2 to 22 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 22 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), `if`(t<0, 0, b(n-1, y+1, 1, 0)*z^h)+ `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+ `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..22); # Alois P. Heinz, Jun 06 2016 -
Mathematica
b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[If[n == 0, (1 - t), If[t < 0, 0, b[n - 1, y + 1, 1, 0]*z^h] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + 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, 22}] // Flatten (* Jean-François Alcover, Nov 29 2016 after Alois P. Heinz *)
Formula
G.f.: G=G(t,z), where t marks number of 1-width valleys and z marks semiperimeter, satisfies z(t+z-tz)G^2-(1-2z-z^2-z^3+tz^3)G+z^2 = 0.
For p-width valleys, the corresponding g.f. satisfies aG^2 - bG + c = 0, where a = z-(1-t)(1-z)z^p, b = 1-2z-z^2(1+z^p-tz^p), c=z^2.
Comments