A273345 -id:A273345 - cp's OEIS frontend

A273344 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k levels. A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step.

1, 1, 1, 3, 2, 6, 7, 14, 19, 2, 33, 53, 11, 79, 148, 47, 1, 194, 409, 181, 10, 482, 1137, 639, 69, 1214, 3159, 2166, 360, 6, 3090, 8793, 7110, 1646, 66, 7936, 24515, 22831, 6868, 490, 2, 20544, 68443, 72145, 26893, 2918, 44, 53545, 191367, 225138, 100598, 15085, 486, 140399, 535762, 695798, 363360, 70847, 3825

Offset: 2

Emeric Deutsch, May 21 2016

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

			Row 4 is 3,2 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] having 1, 0, 0, 1, 0 levels, respectively.
Triangle starts
1;
1,1;
3,2;
6,7;
14,19,2.

Alois P. Heinz, Rows n = 2..250, flattened
A. Blecher, C. Brennan, and A. Knopfmacher, Levels in bargraphs, Ars Math. Contemp., 9, 2015, 297-310.
A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.

Cf. A025243, A082582, A273345.

G := (1-2*z-z^2+2*z^3-2*t*z^3-sqrt((1-z)*(1-3*z-z^2+3*z^3-4*t*z^3+4*z^4 -4*t*z^4-4*z^5+8*t*z^5-4*t^2*z^5)))/(2*z*(1-z+t*z)): 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
# second Maple program:
b:= proc(n, y, t, w) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1, 0))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1, 0))+ `if`(y<1, 0,
      `if`(w=1, z, 1)*b(n-1, y, 0, min(w+1, 2)))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$3)):
seq(T(n), n=2..20);  # Alois P. Heinz, Jun 04 2016

b[n_, y_, t_, w_] := b[n, y, t, w] = Expand[If[n == 0, (1 - t), If[t < 0, 0, b[n - 1, y + 1, 1, 0]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1, 0]] + If[y < 1, 0, If[w == 1, z, 1]*b[n - 1, y, 0, Min[w + 1, 2]]]]]; 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 *)

T(n,0) = A025243(n+1).
Sum(k*T(n,k), k>=1) = A273345(n).
G.f.: G(t,z) = (1-2z-z^2+2z^3-2tz^3-sqrt((1-z)(1-3z-z^2+3z^3-4tz^3+4z^4-4tz^4-4z^5+8tz^5-4t^2z^5)))/(2z(1-z+tz)); z marks semiperimeter, t marks levels. See eq. (2.4) in the Blecher et al. Ars. Math. Contemp. reference (set x = z, y = z, w = t).

A277973 Sum of horizontal positions of the first peak in all bargraphs of semiperimeter n.

Original entry on oeis.org

0, 0, 0, 1, 6, 25, 91, 311, 1029, 3346, 10778, 34544, 110444, 352785, 1126885, 3601617, 11521648, 36899528, 118322448, 379908707, 1221423149, 3932113059, 12675055399, 40909511880, 132200481507, 427718677728, 1385419058692, 4492446685542, 14582927712740, 47385785436719

Offset: 1

Arnold Knopfmacher, Nov 07 2016

nonn

Horizontal position is x-coordinate of the start of the leftmost horizontal step of the first peak.

			For n = 4, a(4) = 1, as only the bargraph with first column of height one and second column of height two has horizontal position 1, all other cases are zero.

Andrew Howroyd, Table of n, a(n) for n = 1..500
A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Cf. A271941, A273720, A273345, A277999.

seq(n) = my(r=sqrt((1 - x)*(1 - 3*x - x^2 - x^3) + O(x^(n-2)))); Vec(2*x^3*(1 + x^2 - r) / ((1 - x)*(1 - 2*x - x^2 + r)^2), -n) \\ Andrew Howroyd, Jan 12 2024

G.f.: (2*x^3*(x^2-sqrt(x^4+2*x^2-4*x+1)+1)) / ((1-x)*(-x^2+sqrt(x^4+2*x^2-4*x+1)-2*x+1)^2).

A274486 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal segments (n>=2, k>=1). A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0).

Original entry on oeis.org

1, 2, 3, 2, 4, 8, 1, 5, 20, 10, 6, 40, 45, 6, 7, 70, 140, 56, 2, 8, 112, 350, 280, 44, 9, 168, 756, 1008, 366, 20, 10, 240, 1470, 2940, 1920, 320, 5, 11, 330, 2640, 7392, 7590, 2552, 190, 12, 440, 4455, 16632, 24684, 13904, 2445, 70

Offset: 2

Emeric Deutsch and Sergi Elizalde, Jun 27 2016

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

Sum(k*T(n,k), k>=0) = A273345(n+1).

			Row 4 is 3,2 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 1,2,2,1,1 horizontal segments.
Triangle starts
1;
2;
3,2;
4,8,1;
5,20,10;
6,40,45,6.

Alois P. Heinz, Rows n = 2..200, 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
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A082582, A273345.

G := ((1-2*z+z^2-2*t*z^2-sqrt((1-z)*((1-z)^3-4*t*z^2*(1-z+t*z))))*(1/2))/(t*z): Gser := simplify(series(G,z = 0,23)): for n from 2 to 18 do P[n] := sort(expand(coeff(Gser,z,n))) end do: for n from 2 to 18 do seq(coeff(P[n],t,j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
# second Maple program:
b:= proc(n, y, t) option remember; expand(
      `if`(n=0, (1-t), `if`(t<0, 0, b(n-1, y+1, 1))+
      `if`(t>0 or y<2, 0, b(n, y-1, -1))+
      `if`(y<1, 0, b(n-1, y, 0)*`if`(t=0, 1, z))))
    end:
T:= n-> (p-> seq(coeff(p, z, i), i=1..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..20);  # Alois P. Heinz, Jun 27 2016

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]*If[t == 0, 1, z]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 20}] // Flatten (* Jean-François Alcover, Dec 02 2016, after Alois P. Heinz *)

G.f.: G = (1-2z+z^2-2tz^2-sqrt((1-z)((1-z)^3-4tz^2*(1-z+tz))))/(2tz).

A277999 Sum of distances between leftmost and rightmost peaks in all bargraphs of semiperimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 9, 53, 261, 1165, 4887, 19642, 76519, 291095, 1086946, 3998430, 14530223, 52272218, 186467253, 660449671, 2325124444, 8143334776, 28393762841, 98621419068, 341403900888, 1178425064256, 4057244213071, 13937739553781, 47786215201214, 163554669548711

Offset: 1

Arnold Knopfmacher, Nov 08 2016

nonn

			a(6)=1 since the bargraph with column heights 2,1,2 has a distance of 1 between first and last peak. All other bargraphs of semiperimeter 6 have at most one peak, hence 0 difference.

A. Blecher, C. Brennan, and A. Knopfmacher, Peaks in bargraphs, Trans. Royal Soc. South Africa, 71, No. 1, 2016, 97-103.

Cf. A271941, A273720, A273345, A277973.

my(x = 'x + O('x^30)); sqx = sqrt(x^4+2*x^2-4*x+1); concat(vector(5), Vec(-(4*x^6*(3-2*x^3+3*x^4 - sqx + x^2*(4-3*sqx) + 2*x*(sqx - 4))/((x^2-3*x+1)*sqx*(-1+2*x+x^2-sqx)^3)))) \\ Michel Marcus, Feb 25 2019

G.f.: -(4*x^6*(3-2*x^3+3*x^4 - sqx + x^2*(4-3*sqx) + 2*x*(sqx - 4))/((x^2-3*x+1)*sqx*(-1+2*x+x^2-sqx)^3)) where sqx = sqrt(x^4+2*x^2-4*x+1).

cp's OEIS Frontend

A273344 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k levels. A level in a bargraph is a maximal sequence of two or more adjacent horizontal steps; it is preceded and followed by either an up step or a down step.

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A277973 Sum of horizontal positions of the first peak in all bargraphs of semiperimeter n.

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A274486 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having k horizontal segments (n>=2, k>=1). A horizontal segment is a maximal sequence of adjacent horizontal steps (1,0).

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Maple

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A277999 Sum of distances between leftmost and rightmost peaks in all bargraphs of semiperimeter n.

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Examples

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Crossrefs

Programs

PARI

Formula