A273342 - OEIS

%I #17 Aug 19 2017 23:03:30

%S 1,1,1,2,2,1,5,4,3,1,13,10,7,4,1,35,27,18,11,5,1,97,75,50,30,16,6,1,

%T 275,213,143,86,47,22,7,1,794,616,416,253,140,70,29,8,1,2327,1808,

%U 1227,754,424,218,100,37,9,1,6905,5372,3661,2269,1295,681,327,138,46,10,1,20705,16127,11030,6885,3978,2133,1056,475,185,56,11,1

%N Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having length of first column k (n>=2, k>=1).

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

%C Sum(k*T(n,k), k>=1) = A273343(n).

%H Alois P. Heinz, <a href="/A273342/b273342.txt">Rows n = 2..150, 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(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).

%e 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.

%p 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

%p # second Maple program:

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

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

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

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

%p     end:

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

%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, 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 T[n_] := Function [p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][ b[n, 0, 0, 1]];

%t Table[T[n], {n, 2, 20}] // Flatten (* _Jean-François Alcover_, Jul 29 2016, after _Alois P. Heinz_ *)

%Y Cf. A082582, A273343.

%K nonn,tabl

%O 2,4

%A _Emeric Deutsch_, May 21 2016