A276066 - OEIS

%I #18 Aug 19 2017 23:08:41

%S 1,1,0,1,1,2,1,0,1,2,4,1,4,1,0,1,4,6,8,8,1,6,1,0,1,7,14,22,12,19,12,1,

%T 8,1,0,1,13,34,43,48,55,18,35,16,1,10,1,0,1,26,72,105,148,109,116,103,

%U 24,56,20,1,12,1,0,1,52,154,276,344,347,398,205,232,166,30,82,24,1,14,1,0,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps.

%C Number of entries in row n is 2n-3.

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

%C T(n,0) = A023431(n-2) = A025246(n+1).

%C Sum(k*T(n,k),k>=0) = 2*A273714(n).

%H Alois P. Heinz, <a href="/A276066/b276066.txt">Rows n = 2..140, flattened</a>

%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 zG^2 - (1-z - t^2*z - 2tz^2+t^2*z^2)G + z^2 = 0.

%F The g.f. B(t,s,z) of bargraphs, where t(s) marks double rises (falls) and z marks semiperimeter, satisfies zB^2 - (1-(1+ts)z +(ts- t-s)z^2)B + z^2 = 0.

%e Row 4 is 1,2,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that they have a total of  0, 1, 1, 2, 4 double rises and double falls, respectively.

%e Triangle starts:

%e 1;

%e 1,0,1;

%e 1,2,1,0,1;

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

%e 4,6,8,8,1,6,1,0,1.

%p eq := z*G^2-(1-z-t^2*z-2*t*z^2+t^2*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. 2*n-4) 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       z^(y-1), `if`(t<0, 0, b(n-1, y+1, 1)*`if`(t=1, z, 1))+

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

%p      `if`(y<1, 0, b(n-1, y, 0))))

%p     end:

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

%p seq(T(n), n=2..12);  # _Alois P. Heinz_, Aug 25 2016

%t b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, (1 - t)*z^(y - 1), If[t < 0, 0, b[n - 1, y + 1, 1]*If[t == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]*If[t == -1, z, 1]] + If[y < 1, 0, b[n - 1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 12}] // Flatten (* _Jean-François Alcover_, Dec 02 2016 after _Alois P. Heinz_ *)

%Y Cf. A023431, A025246, A082582, A273714.

%K nonn,tabf

%O 2,6

%A _Emeric Deutsch_ and _Sergi Elizalde_, Aug 25 2016