A161680 -id:A161680 - cp's OEIS frontend

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 13, 10, 1, 1, 9, 22, 25, 15, 1, 1, 11, 33, 46, 41, 21, 1, 1, 13, 46, 73, 79, 61, 28, 1, 1, 15, 61, 106, 129, 121, 85, 36, 1, 1, 17, 78, 145, 191, 201, 172, 113, 45, 1, 1, 19, 97, 190, 265, 301, 289, 232, 145, 55, 1, 1, 21

Offset: 0

Philippe Deléham, Mar 16 2014

			Square array begins:
n\k : 0......1......2......3......4......5......6......7......8......9
======================================================================
.0||  1......1......1......1......1......1......1......1......1......1
.1||  1......3......5......7......9.....11.....13.....15.....17.....19
.2||  1......6.....13.....22.....33.....46.....61.....78.....97....118
.3||  1.....10.....25.....46.....73....106....145....190....241....298
.4||  1.....15.....41.....79....129....191....265....351....449....559
.5||  1.....21.....61....121....201....301....421....561....721....901
.6||  1.....28.....85....172....289....436....613....820...1057...1324
.7||  1.....36....113....232....393....596....841...1128...1457...1828
.8||  1.....45....145....301....513....781...1105...1485...1921...2413
.9||  1.....55....181....379....649....991...1405...1891...2449...3079
10||  1.....66....221....466....801...1226...1741...2346...3041...3826
11||  1.....78....265....562....969...1486...2113...2850...3697...4654

Cf. Columns: A000012, A000217, A001844, A038764, A081585, A081587, A081589, A081591, A081593.

Cf. Rows: A000012, A005408, A028872, A100536, A239325, A069133.

T(n,k) = 3*T(n-1,k) - 3*T(n-2,k) + T(n-3,k).
T(n,k) = 3*T(n,k-1) - 3*T(n,k-2) + T(n,k-3).
T(n,k) = (T(n,k-1) + T(n,k+1))/2 - A161680(n).
T(n,k) = (T(n-1,k) + T(n+1,k) - A000290(n))/2.

A271024 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i < j such that i and j are in different blocks; triangle T(n,k), n >= 0, 0 <= k <= n*(n-1)/2 read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 3, 1, 1, 0, 0, 4, 3, 6, 1, 1, 0, 0, 0, 5, 0, 10, 10, 15, 10, 1, 1, 0, 0, 0, 0, 6, 0, 0, 15, 25, 0, 60, 35, 45, 15, 1, 1, 0, 0, 0, 0, 0, 7, 0, 0, 0, 21, 21, 35, 0, 105, 105, 105, 210, 140, 105, 21, 1, 1, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 28, 28

Offset: 0

Alois P. Heinz, Mar 28 2016

			T(3,0) = 1: 123.
T(3,2) = 3: 12|3, 13|2, 1|23.
T(3,3) = 1: 1|2|3.
Triangle T(n,k) begins:
  1;
  1;
  1, 1;
  1, 0, 3, 1;
  1, 0, 0, 4, 3, 6,  1;
  1, 0, 0, 0, 5, 0, 10, 10, 15, 10, 1;
  1, 0, 0, 0, 0, 6,  0,  0, 15, 25, 0, 60, 35, 45, 15, 1;

Alois P. Heinz, Rows n = 0..40, flattened
Wikipedia, Partition of a set

Row sums give A000110.

Cf. A161680, A271023.

b:= proc(n, l) option remember; `if`(n=0, x^(m->
      add(j*(m-j)/2, j=l))(add(i, i=l)), b(n-1, [l[], 1])+
      add(b(n-1, subsop(j=l[j]+1, l)), j=1..nops(l)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, [])):
seq(T(n), n=0..10);

b[n_, l_] := b[n, l] = If[n == 0, x^Function[m, Sum[(1/2)*j*(m - j), {j, l}]][Total[l]], Sum[b[n - 1, ReplacePart[l, j -> l[[j]] + 1]], {j, 1, Length[l]}] + b[n - 1, Append[l, 1]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, {}]];
Flatten[Table[T[n], {n, 0, 10}]] (* Jean-François Alcover, May 27 2018, translated from Maple *)

T(n,k) = A271023(n,n*(n-1)/2-k).

T(n,n-1) = n for n >= 3.

A337206 Cardinality of maximal level sets of Gini index on integer partitions.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 5, 5, 7, 8, 9, 11, 13, 15, 17, 21, 23, 28, 33, 38, 44, 52, 60, 72, 81, 95, 112, 128, 147, 175, 195, 233, 267, 305, 353, 412, 462, 533, 617, 703, 807, 932, 1052, 1210, 1389, 1569, 1785, 2060, 2315, 2642, 3023, 3405, 3876, 4413, 4968

Offset: 0

Grant Kopitzke, Aug 18 2020

nonn

a(n) is a lower bound on A076269(n).

			For n=6 the maximal level set of the Gini index contains the partitions (3,3) and (4,1,1). So a(6)=2.

Alois P. Heinz, Table of n, a(n) for n = 0..300
Grant Kopitzke, The Gini Index of an Integer Partition, arXiv:2005.04284 [math.CO], 2020.

Lower bound on A076269.

Cf. A161680, A264034.

b:= proc(n, i, w) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, w)+expand(x^(w*i)*b(n-i, min(i, n-i), w+1))))
    end:
a:= n-> max(coeffs(b(n$2, 0))):
seq(a(n), n=0..61);  # Alois P. Heinz, Jan 20 2023

m = 75;
p = Product[ 1/(1 - q^Binomial[i + 1, 2] x^i), {i, 1, m}];
psn = Expand@Normal@Series[ p, {x, 0, m}];
psnc = CoefficientList[CoefficientList[psn, {x}, {m}], {q}];
Map[Max, psnc]

G.f.: Product_{n=1..oo} 1/(1-q^(binomial(n+1,2))x^n)-1 = Sum_{n=1..oo} Sum_{lambda a partition of n} q^(binomial(n+1,2)-g(lambda))x^n, where g(lambda) is the Gini index of lambda.

a(n) = max_{k=0..A161680(n)} A264034(n,k). - Alois P. Heinz, Jan 20 2023

Typo in a(43) corrected by Alois P. Heinz, Jan 20 2023

A342939 a(n) is the Skolem number of the triangular grid graph T_n.

Original entry on oeis.org

1, 2, 5, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 277, 301, 326, 352, 379, 407, 436, 466, 497, 529, 562, 596, 631, 667, 704, 742, 781, 821, 862, 904, 947, 991, 1036, 1082, 1129, 1177, 1226, 1276, 1327, 1379, 1432, 1486

Offset: 1

Stefano Spezia, Mar 30 2021

For the meaning of Skolem number of a graph, see Definitions 1.4 and 1.5 in Carrigan and Green.

Braxton Carrigan and Garrett Green, Skolem Number of Subgraphs on the Triangular Lattice, Communications on Number Theory and Combinatorial Theory 2 (2021), Article 2.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A152947, A161680, A247476, A342938, A342940.

For n > 1, 3*A002061(n) gives the Skolem number of the hexagonal grid graph H_n.

LinearRecurrence[{3,-3,1},{1,2,5,7,11,16},55]

O.g.f.: x*(1 - x + 2*x^2 - 3*x^3 + 3*x^4 - x^5)/(1 - x)^3.
E.g.f.: exp(x)*(2 + x^2)/2 - 1 + x^3/6.
a(n) = 3*a(n-1) - 3*a(n-2) - a(n-3) for n > 6.
Except for a(3) = 5:
  a(n) = 1 + n*(n - 1)/2 (see Theorem 2.5 in Carrigan and Green).
  a(n) = 1 + A161680(n).
  a(n) = A152947(n-1).

A350295 2nd subdiagonal of the triangle A350292.

Original entry on oeis.org

6, 8, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1225, 1275, 1326, 1378, 1431, 1485, 1540

Offset: 3

Stefano Spezia, Dec 23 2021

Harvey P. Dale, Table of n, a(n) for n = 3..1000
Heiko Harborth and Hauke Nienborg, Saturated vertex Turán numbers for cube graphs, Congr. Num. 208 (2011), 183-188.
Mathonline, Cube Graphs
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A112355, A161680, A350292.

Join[{6,8},Table[Binomial[n,2],{n,5,56}]]
LinearRecurrence[{3,-3,1},{6,8,10,15,21},60] (* Harvey P. Dale, Jul 01 2022 *)

a(n) = binomial(n, 2) = A000217(n-1) for n > 4 with a(3) = 6 and a(4) = 8 (see Theorem 3 in Harborth and Nienborg).
O.g.f.: x^3*(2*x^4 - 3*x^3 - 4*x^2 + 10*x - 6)/(x - 1)^3.
E.g.f.: x^2*(x^2 + 6*x + 6*exp(x) - 6)/12.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 7.

A362977 The x-coordinates of the even-ranked elements of the lexicographically earliest sequence of points satisfying staircase and no-three-in-a-line conditions (see comments).

Original entry on oeis.org

0, 1, 2, 5, 6, 3, 12, 13, 4, 15, 8, 27, 19, 11, 9, 32, 25, 16, 33, 35, 59, 30, 24, 7, 51, 45, 10, 18, 29, 26, 54, 58, 14, 53, 23, 34, 37, 49, 44, 60, 43, 31, 41, 66, 17, 72, 80, 81, 78, 88, 87, 46, 83, 22, 76, 79, 130, 97, 108, 111, 94, 119, 153, 89, 39, 112

Offset: 0

Luc Rousseau, May 11 2023

nonn

We construct the lexicographically earliest sequence of points M(i), for i >= 0, with the following rules:
- their (x, y) coordinates are taken among the nonnegative integers;
- if i > 0 and i is odd, then "move horizontally to a free column"; i.e., M(i) must have the same y as M(i-1), and M(i) is not allowed to have the same x as any M(k) for k < i;
- if i > 0 and i is even, then "move vertically to a free row"; i.e., M(i) must have the same x as M(i-1), and M(i) is not allowed to have the same y as any M(k) for k < i;
- three points are not allowed to be aligned.
See SVG illustration, Links section.
Then a(n) (resp. A362978(n)) is defined as the x-coordinate (resp. y-coordinate) of M(i), where i := 2n (to eliminate duplicates).

			  y
  ^
  | . . . 9 . . 8
  | . . 4 . . 5 .
  | . . . . . 6 7
  | . 2 3 . . . .
  | 0 1 . . . . .
  +-------------------> x
    0 1 2 3 4 5 6
Abscissas of the points 0, 2, 4, 6, 8, ...: 0, 1, 2, 5, 6, ...

Luc Rousseau, SVG illustration, n = 0..126
Luc Rousseau, Java program for the combined computation of A362977 and A362978.

Cf. A362978 (the corresponding y-coordinates).

Cf. A161680 (number of nonalignment checks to pass).

Java
```
// See Rousseau link.
```

A366132 Number of unordered pairs of distinct strict integer partitions of n.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 6, 10, 15, 28, 45, 66, 105, 153, 231, 351, 496, 703, 1035, 1431, 2016, 2850, 3916, 5356, 7381, 10011, 13530, 18336, 24531, 32640, 43660, 57630, 75855, 100128, 130816, 170820, 222778, 288420, 372816, 481671, 618828, 793170, 1016025, 1295245

Offset: 0

Gus Wiseman, Oct 08 2023

nonn

			The a(3) = 1 through a(8) = 15 pairs of strict partitions:
  {3,21}  {4,31}  {5,32}   {6,42}    {7,43}    {8,53}
                  {5,41}   {6,51}    {7,52}    {8,62}
                  {41,32}  {51,42}   {7,61}    {8,71}
                           {6,321}   {52,43}   {62,53}
                           {42,321}  {61,43}   {71,53}
                           {51,321}  {61,52}   {71,62}
                                     {7,421}   {8,431}
                                     {43,421}  {8,521}
                                     {52,421}  {53,431}
                                     {61,421}  {53,521}
                                               {62,431}
                                               {62,521}
                                               {71,431}
                                               {71,521}
                                               {521,431}

For subsets instead of partitions we have A006516, non-disjoint A003462.
The disjoint case is A108796, non-strict A260669.
For non-strict partitions we have A355389.
The ordered disjoint case is A365662, non-strict A054440.
The ordered version is 2*a(n).
Including equal pairs or twins gives A366317, ordered A304990.
A000041 counts integer partitions, strict A000009.
A002219 and A237258 count partitions of 2n including a partition of n.
A161680 and A000217 count 2-subsets of {1..n}.
Cf. A000124, A001255, A006827, A032302, A064914, A365661, A365663.

Table[Length[Subsets[Select[IntegerPartitions[n],UnsameQ@@#&],{2}]],{n,0,30}]

a(n) = binomial(A000009(n),2).

A370946 Number of partitions of [n] whose non-singleton elements sum to n.

Original entry on oeis.org

1, 0, 0, 1, 1, 2, 3, 4, 5, 7, 12, 14, 20, 26, 36, 54, 68, 90, 120, 157, 202, 296, 360, 480, 612, 803, 1006, 1317, 1764, 2198, 2821, 3592, 4552, 5754, 7269, 9074, 11990, 14646, 18586, 23112, 29208, 35972, 45277, 55584, 69350, 87881, 107609, 133068, 165038

Offset: 0

Alois P. Heinz, Mar 06 2024

nonn

			a(0) = 1: the empty partition.
a(3) = 1: 12|3.
a(4) = 1: 13|2|4.
a(5) = 2: 1|23|4|5, 14|2|3|5.
a(6) = 3: 123|4|5|6, 1|24|3|5|6, 15|2|3|4|6.
a(7) = 4: 124|3|5|6|7, 1|2|34|5|6|7, 1|25|3|4|6|7, 16|2|3|4|5|7.
a(8) = 5: 125|3|4|6|7|8, 134|2|5|6|7|8, 1|2|35|4|6|7|8, 1|26|3|4|5|7|8, 17|2|3|4|5|6|8.
a(9) = 7: 126|3|4|5|7|8|9, 135|2|4|6|7|8|9, 1|234|5|6|7|8|9, 1|2|3|45|6|7|8|9, 1|2|36|4|5|7|8|9, 1|27|3|4|5|6|8|9, 18|2|3|4|5|6|7|9.
a(10) = 12: 1234|5|6|7|8|9|10, 12|34|5|6|7|8|9|10, 127|3|4|5|6|8|9|10, 13|24|5|6|7|8|9|10, 136|2|4|5|7|8|9|10, 14|23|5|6|7|8|9|10, 1|235|4|6|7|8|9|10, 145|2|3|6|7|8|9|10, 1|2|3|46|5|7|8|9|10, 1|2|37|4|5|6|8|9|10, 1|28|3|4|5|6|7|9|10, 19|2|3|4|5|6|7|8|10.

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Wikipedia, Partition of a set

Cf. A161680, A370945.

h:= proc(n) option remember; `if`(n=0, 1,
      add(h(n-j)*binomial(n-1, j-1), j=2..n))
    end:
b:= proc(n, i, m) option remember; `if`(n>i*(i+1)/2, 0,
     `if`(n=0, h(m), b(n, i-1, m)+b(n-i, min(n-i, i-1), m+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..48);

h[n_] := h[n] = If[n == 0, 1, Sum[h[n-j]*Binomial[n-1, j-1], {j, 2, n}]];
b[n_, i_, m_] := b[n, i, m] = If[n > i*(i + 1)/2, 0, If[n == 0, h[m], b[n, i - 1, m] + b[n - i, Min[n - i, i - 1], m + 1]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 48}] (* Jean-François Alcover, Mar 08 2024, after Alois P. Heinz *)

a(n) = A370945(n,n*(n-1)/2).

A376832 Irregular triangle read by rows: the n-th row gives the number of points of an n X n square lattice that lie above or to the left of a line of increasing slope that passes through two lattice points one of which is the bottom-left corner of the lattice, (0, 0).

Original entry on oeis.org

2, 1, 0, 6, 5, 3, 2, 0, 12, 11, 10, 9, 6, 5, 4, 3, 0, 20, 19, 18, 17, 16, 15, 14, 10, 9, 8, 7, 6, 5, 4, 0, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 0, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 0

Offset: 2

Stefano Spezia, Dec 22 2024

The increasing slopes of the line are given by the Farey series of order n - 1. Specifically, they are given by the fractions A006842(n-1)/A006843(n-1) followed by their reciprocals A006843(n-1)/A006842(n-1) in reverse order, with the fraction 1/1 included only once.

			The irregular triangle begins as:
   2,  1,  0;
   6,  5,  3,  2,  0;
  12, 11, 10,  9,  6,  5,  4,  3, 0;
  20, 19, 18, 17, 16, 15, 14, 10, 9, 8, 7, 6, 5, 4, 0;
  ...

Stefano Spezia, Table of n, a(n) for n = 2..9081 (first 30 rows of the irregular triangle)

Cf. A002378, A006842, A006843, A118403 (row lengths), A161680, A379540 (row sums).

A118403[n_]:=SeriesCoefficient[(1-2*x+2*x^2)*(1+x^2)/(1-x)^3,{x,0,n}]; T[n_,k_]:=If[1<=k<(A118403[n]+1)/2,n(n-1)-k+1,If[(A118403[n]+1)/2<=k<A118403[n],n(n-1)/2-k+(A118403[n]+1)/2,0]]; Table[T[n,k],{n,2,7},{k,A118403[n]}]//Flatten

T(n, k) = n*(n - 1) - k + 1 for 1 <= k < (A118403(n)+1)/2.
T(n, k) = n*(n - 1)/2 - k + (A118403(n)+1)/2 for (A118403(n)+1)/2 <= k < A118403(n).
T(n, A118403(n)) = 0.

A110195 a(n) = 11^((n^2-n)/2).

Original entry on oeis.org

1, 1, 11, 1331, 1771561, 25937424601, 4177248169415651, 7400249944258160101211, 144209936106499234037676064081, 30912680532870672635673352936887453361, 72890483685103052142902866787761839379440139451, 1890591424712781041871514584574319778449301246603238034051

Offset: 0

Philippe Deléham, Sep 07 2005

Sequence given by the Hankel transform (see A001906 for definition) of A082173 = {1, 1, 12, 155, 2124, 30482, 453432, 6936799, ...}; example : det([1, 1, 12, 155; 1, 12, 155, 2124; 12, 155, 2124, 30482; 155, 2124, 30482, 453432]) = 11^6 = 1771561.

Cf. A001020, A006125, A047656, A053763, A053764, A109345, A109354, A109493, A109966, A110147, A161680.

Table[11^((n^2-n)/2),{n,0,20}] (* Harvey P. Dale, Feb 02 2012 *)
Join[{1,1},Table[Det[Table[Binomial[11i,j],{i,n},{j,n}]],{n,10}]] (* Harvey P. Dale, Apr 01 2019 *)

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(11i, j).

a(n) = A001020(A161680(n)).

a(11) from Harvey P. Dale, Feb 02 2012

a(12) from Jason Yuen, Aug 29 2025

cp's OEIS Frontend

A239331 Square array, read by antidiagonals: column k has g.f. (1+(k-1)*x)^2/(1-x)^3.

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A271024 Number T(n,k) of set partitions of [n] having exactly k pairs (i,j) with i < j such that i and j are in different blocks; triangle T(n,k), n >= 0, 0 <= k <= n*(n-1)/2 read by rows.

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Maple

Mathematica

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A337206 Cardinality of maximal level sets of Gini index on integer partitions.

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Mathematica

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Extensions

A342939 a(n) is the Skolem number of the triangular grid graph T_n.

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Mathematica

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A350295 2nd subdiagonal of the triangle A350292.

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A362977 The x-coordinates of the even-ranked elements of the lexicographically earliest sequence of points satisfying staircase and no-three-in-a-line conditions (see comments).

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Comments

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Java

A366132 Number of unordered pairs of distinct strict integer partitions of n.

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Examples

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Programs

Mathematica

Formula

A370946 Number of partitions of [n] whose non-singleton elements sum to n.

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Maple

Mathematica