A124287 -id:A124287 - cp's OEIS frontend

A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 3, 2, 2, 1, 1, 3, 4, 4, 3, 2, 1, 1, 2, 5, 5, 4, 3, 2, 1, 1, 4, 7, 8, 6, 5, 3, 2, 1, 1, 3, 8, 9, 9, 6, 5, 3, 2, 1, 1, 5, 11, 14, 12, 10, 7, 5, 3, 2, 1, 1, 4, 12, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 7, 16, 23, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 5, 18, 25, 28, 24, 20, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 3

T. D. Noe, Oct 24 2006

For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides.

T(n,k) = number of partitions of n into k parts (k >= 3) in which all parts are less than n/2. Also T(n,k) = number of partitions of 2*n into k parts in which all parts are even and less than n. - L. Edson Jeffery, Mar 19 2012

			For polygons having perimeter 7: 2 triangles, 2 quadrilaterals, 2 pentagons, 1 hexagon and 1 heptagon. The triangle begins
1
0 1
1 1 1
1 1 1 1
2 2 2 1 1
1 3 2 2 1 1

T. D. Noe, Rows n=3..102 of triangle, flattened
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis IX: k-Gon Partitions, Bull. Austral. Math. Soc. 64 (2001), 321-329.
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A124287 (similar, but with no restriction on the sides).
Cf. A210249 (gives row sums of this sequence for n >= 3).
Cf. A005044 (k=3), A062890 (k=4), A069906 (k=5), A069907 (k=6), A288253 (k=7), A288254 (k=8), A288255 (k=9), A288256 (k=10).

b:= proc(n, i) option remember; `if`(n=0, [1], `if`(i<1, [],
      zip((x, y)-> x+y, b(n, i-1), `if`(i>n, [],
                    [0, b(n-i, i)[]]), 0)))
    end:
T:= n-> b(n, ceil(n/2)-1)[4..n+1][]:
seq(T(n), n=3..20);  # Alois P. Heinz, Jul 15 2013

Flatten[Table[p=IntegerPartitions[n]; Length[Select[p, Length[ # ]==k && #[[1]] < Total[Rest[ # ]]&]], {n,3,30}, {k,3,n}]]
(* second program: *)
QP = QPochhammer; T[n_, k_] := SeriesCoefficient[x^k*(1/QP[x, x, k] + x^(k - 2)/((x-1)*QP[x^2, x^2, k-1])), {x, 0, n}]; Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jan 08 2016 *)

G.f. for column k is x^k/(product_{i=1..k} 1-x^i) - x^(2k-2)/(1-x)/(product_{i=1..k-1} 1-x^(2i)).

A293819 Triangle read by rows of the number of integer-sided k-gons having perimeter n, modulo rotations but not reflections, for k=3..n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 3, 1, 1, 1, 6, 6, 4, 1, 1, 4, 10, 13, 10, 4, 1, 1, 2, 12, 21, 21, 12, 5, 1, 1, 5, 20, 37, 41, 30, 15, 5, 1, 1, 4, 23, 51, 74, 65, 43, 19, 6, 1, 1, 7, 35, 84, 126, 131, 99, 55, 22, 6, 1, 1, 5, 38, 108, 196, 239, 216, 143, 73, 26, 7, 1, 1, 10, 56, 166, 314, 422, 428

Offset: 3

James East, Oct 16 2017

Rotations are counted only once, but reflections are considered different. For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2). Column k=3 is A008742, column k=4 is A293821, column k=5 is A293822 and column k=6 is A293823.

A formula is given in Section 6 of the East and Niles article.

			For polygons having perimeter 7, there are: 2 triangles (331, 322), 4 quadrilaterals (3211, 3121, 3112, 2221), 3 pentagons (31111, 22111, 21211), 1 hexagon (211111) and 1 heptagon (1111111). Note that the quadrilaterals 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.
The triangle begins:
n=3:  1;
n=4:  0,  1;
n=5:  1,  1,  1;
n=6:  1,  2,  1,  1;
n=7:  2,  4,  3,  1,  1;
n=8:  1,  6,  6,  4,  1,  1;
n=9:  4, 10, 13, 10,  4,  1,  1;
...

Andrew Howroyd, Rows n=3..52 of triangle, flattened
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Columns: A008742 (triangles), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons).
Row sums are A293820.
Same triangle with reflection allowed is A124287.

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#]&]/n - Binomial[Floor[n/2], k - 1];
Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jun 14 2018, translated from PARI *)

T(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial(n/d,k/d))/n - binomial(floor(n/2), k-1)}
for(n=3, 10, for(k=3, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 21 2017

T(n,k) = (Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d))/n - binomial(floor(n/2), k-1). - Andrew Howroyd, Nov 21 2017

A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 32, 54, 102, 180, 336, 607, 1144, 2098, 3960, 7397, 14022, 26452, 50404, 95821, 183322, 350554, 673044, 1292634, 2489502, 4797694, 9264396, 17904220, 34652962, 67125898, 130182972, 252679320, 490918440, 954505718, 1857413460, 3616951513, 7048412792, 13744169104

Offset: 3

James East, Oct 16 2017

nonn

Rotations and reversals are counted only once. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. These are row sums of A124287.
A formula is proved in Theorem 1.6 of the East and Niles article.
The same article shows that a(n) is asymptotic to 2^(n-1) / n.

			There are 10 polygons having perimeter 7: 2 triangles, 3 quadrilaterals, 3 pentagons, 1 hexagon and 1 heptagon.

Andrew Howroyd, Table of n, a(n) for n = 3..200
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Row sums of A124287 (k-gon triangle).

Cf. A293820 (polygons modulo rotations only).

a[n_] := DivisorSum[n, EulerPhi[n/#]*2^# &]/(2*n) + 2^Floor[(n - 3)/2] - If[Mod[n, 4] < 2, 3*2^Floor[(n - 4)/4], 2^Floor[(n + 2)/4] ];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

a(n)={sumdiv(n, d, eulerphi(n/d)*2^d)/(2*n) + 2^floor((n-3)/2) - if(n%4<2, 3*2^floor((n-4)/4), 2^floor((n+2)/4))} \\ Andrew Howroyd, Nov 21 2017

A124285 Number of integer-sided pentagons having perimeter n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 9, 13, 23, 29, 48, 60, 92, 109, 158, 186, 258, 296, 397, 451, 589, 658, 841, 933, 1169, 1283, 1582, 1728, 2100, 2275, 2732, 2948, 3502, 3756, 4419, 4725, 5511, 5866, 6789, 7207, 8283, 8761, 10006, 10560, 11990, 12617, 14250, 14968

Offset: 1

T. D. Noe, Oct 24 2006

nonn

Rotations and reversals are counted only once. Note that this is different from A069906, which counts pentagons whose sides are nondecreasing.

			The three pentagons having perimeter 7 are (1,1,1,2,2), (1,1,2,1,2) and (1,1,1,1,3).

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A057886 (quadrilaterals), A124286 (hexagons), A124287 (k-gons).

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==5 && #[[1]]

Empirical g.f.: -x^5*(x^12 +2*x^9 +2*x^8 +2*x^7 +5*x^6 +3*x^5 +2*x^4 +2*x^3 +x^2 +x +1) / ((x -1)^5*(x +1)^4*(x^2 +1)^2*(x^4 +x^3 +x^2 +x +1)). - Colin Barker, Oct 27 2013

A124286 Number of integer-sided hexagons having perimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 4, 7, 15, 25, 46, 72, 113, 172, 248, 360, 491, 686, 896, 1217, 1536, 2031, 2504, 3236, 3905, 4955, 5880, 7336, 8586, 10556, 12208, 14823, 16964, 20364, 23106, 27456, 30906, 36399, 40692, 47532, 52816, 61237, 67672, 77941, 85701

Offset: 1

T. D. Noe, Oct 24 2006

nonn

Rotations and reversals are counted only once. Note that this is different from A069907, which counts hexagons whose sides are nondecreasing.

			The four hexagons having perimeter 8 are (1,1,1,1,2,2), (1,1,1,2,1,2), (1,1,2,1,1,2) and (1,1,1,1,1,3).

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A057886 (quadrilaterals), A124285 (pentagons), A124287 (k-gons).

Needs["DiscreteMath`Combinatorica`"]; Table[s=Select[Partitions[n], Length[ # ]==6 && #[[1]]

Empirical g.f.: x^6*(x^13 +3*x^12 +6*x^11 +6*x^10 +10*x^9 +9*x^8 +12*x^7 +10*x^6 +8*x^5 +5*x^4 +4*x^3 +2*x^2 +x +1) / ((x -1)^6*(x +1)^5*(x^2 -x +1)*(x^2 +1)^2*(x^2 +x +1)^2). - Colin Barker, Oct 27 2013

A342307 Table read by ascending antidiagonals: T(n, k) is the maximum number of quasi k-gons that are not k-gons in a finite projective plane of order n, with k >= 3.

Original entry on oeis.org

126, 936, 2520, 3780, 41184, 25200, 11160, 287280, 1029600, 151200, 27090, 1294560, 12927600, 18532800, 529200, 57456, 4442760, 90619200, 439538400, 259459200, 846720, 110376, 12640320, 444276000, 4893436800, 12307075200, 2905943040, 0, 196560, 31346784, 1706443200, 34653528000, 222651374400, 295369804800, 26153487360, 0

Offset: 2

Stefano Spezia, Mar 08 2021

			n\k |     3        4         5           6
----+-------------------------------------
  2 |   126     2520     25200      151200 ...
  3 |   936    41184   1029600    18532800 ...
  4 |  3780   287280  12927600   439538400 ...
  5 | 11160  1294560  90619200  4893436800 ...
...

Vladislav Taranchuk, On the number of k-gons in finite projective planes, (2020).

Cf. A000142, A000217, A000794, A001231, A124278, A124287, A293819.

T[n_,k_]:=k!Binomial[k-1,2]Binomial[n^2+n+1,k-1](n-1); Table[T[n-k+3,k],{n,2,9},{k,3,n+1}]//Flatten

T(n, k) = k!*binomial(k - 1, 2)*binomial(n^2 + n + 1, k - 1)*(n - 1).

cp's OEIS Frontend

A124278 Triangle of the number of nondegenerate k-gons having perimeter n and whose sides are nondecreasing, for k=3..n.

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A293819 Triangle read by rows of the number of integer-sided k-gons having perimeter n, modulo rotations but not reflections, for k=3..n.

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A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

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A124285 Number of integer-sided pentagons having perimeter n.

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A124286 Number of integer-sided hexagons having perimeter n.

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A342307 Table read by ascending antidiagonals: T(n, k) is the maximum number of quasi k-gons that are not k-gons in a finite projective plane of order n, with k >= 3.

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