A273715 - OEIS

%I #27 Aug 19 2017 23:10:03

%S 0,1,1,1,2,3,5,8,13,21,1,34,57,6,90,158,27,241,445,107,1,652,1269,396,

%T 10,1780,3655,1404,66,4899,10611,4838,356,1,13581,31002,16344,1700,15,

%U 37893,91048,54429,7482,135,106340,268536,179332,31070,940,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k peaks of width 1 (i.e., UHD configurations, where U=(0,1), H(1,0), D=(0,-1)), (n>=2, k>=0).

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

%C Sum(k*T(n,k),k>=1) = A273716(n).

%H Alois P. Heinz, <a href="/A273715/b273715.txt">Rows n = 2..250, 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. 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(t,z) satisfies z*G^2 - (1-2*z-z^2-z^3+t*z^3)G + z^2*(t+z-t*z) = 0.

%e Row 4 is 2,3 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 0,1,1,0,1 peaks of width 1.

%e Triangle T(n,k) begins:

%e :    0,    1;

%e :    1,    1;

%e :    2,    3;

%e :    5,    8;

%e :   13,   21,   1;

%e :   34,   57,   6;

%e :   90,  158,  27;

%e :  241,  445, 107,  1;

%e :  652, 1269, 396, 10;

%p eq := z*G^2-(1-2*z-z^2-z^3+t*z^3)*G+z^2*(t+z-t*z) = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): 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 .. degree(P[n])) end do; # yields sequence in triangular form

%p # second Maple program:

%p b:= proc(n, y, t, h) option remember; expand(

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

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

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

%p     end:

%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):

%p seq(T(n), n=2..20);  # _Alois P. Heinz_, Jun 06 2016

%t b[n_, y_, t_, h_] := b[n, y, t, h] = Expand[ If[n == 0, (1 - t)*z^h, If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]*z^h] + 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, 20}] // Flatten (* _Jean-François Alcover_, Nov 29 2016 after _Alois P. Heinz_ *)

%Y Cf. A082582, A273716.

%K nonn,tabf

%O 2,5

%A _Emeric Deutsch_, May 28 2016