A273901 - OEIS

%I #22 Aug 19 2017 23:19:39

%S 0,1,1,0,1,1,3,0,1,2,4,6,0,1,4,11,9,10,0,1,8,24,33,16,15,0,1,16,56,80,

%T 76,25,21,0,1,33,128,218,200,150,36,28,0,1,69,297,558,630,420,267,49,

%U 36,0,1,146,688,1445,1776,1515,784,441,64,45,0,1,312,1601,3684,5091,4635,3213,1344,688,81,55,0,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k odd-length columns (n>=2, k>=0).

%C Number of entries in row n is n.

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

%C T(n,0) = A004149(n-1).

%C Sum(k*T(n,k), k>=0) = A273902(n).

%H M. Bousquet-Mélou and A. Rechnitzer, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00553-5">The site-perimeter of bargraphs</a>, Adv. in Appl. Math. 31 (2003), 86-112.

%H Emeric Deutsch, S Elizalde, <a href="http://arxiv.org/abs/1609.00088">Statistics on bargraphs viewed as cornerless Motzkin paths</a>, arXiv preprint arXiv:1609.00088, 2016

%F G.f.: G = G(t,z) satisfies aG^2 + bG + c = 0, where a = z(1+tz^2), b = tz^3 + z^2 + z^3 + tz^4 - tz^2 + tz + z - 1, c = z^2*(tz^2 + z - tz + t).

%F The trivariate g.f. G(t,s,z), where t (s) marks number of odd-length (even-length) columns and z marks semiperimeter, satisfies AG^2 + BG + C = 0, where A = z(tsz^2 - tsz + tz +s), B = tsz^4+(t+s)z^3+(1-ts)z^2+(t+s)z-1, C = tsz^4+s(1-t)z^3+tz^2.

%e Row 4 is 1,3,0,1 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, they have 3, 1, 1, 0, 1 columns of odd length.

%e Triangle starts

%e 0,1;

%e 1,0,1;

%e 1,3,0,1;

%e 2,4,6,0,1;

%e 4,11,9,10,0,1

%p eq := z*(1+t*z^2)*f^2-(1-z-t*z+t*z^2-t*z^4-z^3-z^2-t*z^3)*f+z^2*(t*z^2+z-t*z+t) = 0: f:= RootOf(eq,f): fser:=simplify(series(f, z = 0,15)): for n from 2 to 12 do P[n] := sort(coeff(fser, z, n)) end do: for n from 2 to 12 do seq(coeff(P[n],t,k),k=0..n-1)end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, y, t) option remember; expand(`if`(n=0, 1-t,

%p       `if`(t<0, 0, b(n-1, y+1, 1))+ `if`(t>0 or y<2, 0,

%p        b(n, y-1, -1))+ `if`(y<1, 0, b(n-1, y, 0)*z^

%p       `if`(y::odd, 1, 0))))

%p     end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..n-1))(b(n, 0$2)):

%p seq(lprint(T(n)), n=2..15);  # _Alois P. Heinz_, Jun 24 2016

%t b[n_, y_, t_] := b[n, y, t] = Expand[If[n==0, 1-t, If[t<0, 0, b[n-1, y+1, 1]] + If[t>0 || y<2, 0, b[n, y-1, -1]] + If[y<1, 0, b[n-1, y, 0]*z^If[OddQ[y], 1, 0]]]]; T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 0, n-1}]][b[n, 0, 0]]; Table[T[n], {n, 2, 15}] // Flatten (* _Jean-François Alcover_, Nov 29 2016 after _Alois P. Heinz_ *)

%Y Cf. A004149, A082582, A273902, A273903, A273904.

%K nonn,tabf

%O 2,7

%A _Emeric Deutsch_, Jun 22 2016