OFFSET
2,4
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
FORMULA
G.f.: G(x,z) satisfies (1 - t - tz^2 + t^2 z)G^2 - t(1 - z)(1- z - tz - tz^2)G + t^2 z^2 (1 - z) = 0 (z marks semiperimeter, x marks length of first column).
EXAMPLE
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] which, clearly, have first-column lengths 1, 1, 2, 2, 3.
MAPLE
eq := (1-t-t*z^2+t^2*z)*G^2-t*(1-z)*(1-z-t*z-t*z^2)*G+t^2*z^2*(1-z) = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 23)): for n from 2 to 20 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 1 .. n-1) 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, h))+
`if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+
`if`(y<1, 0, b(n-1, y, 0, 0)*`if`(h=1, z^y, 1))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..degree(p)))(b(n, 0$2, 1)):
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, If[t < 0, 0, b[n - 1, y + 1, 1, h]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, b[n - 1, y, 0, 0]*If[h == 1, z^y, 1]]]];
T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][ b[n, 0, 0, 1]];
Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Jul 29 2016, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, May 21 2016
STATUS
approved