A052810 -id:A052810 - cp's OEIS frontend

A327029 T(n, k) = Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. Triangle read by rows for 0 <= k <= n.

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 1, 1, 0, 5, 2, 2, 1, 1, 0, 6, 6, 4, 2, 1, 1, 0, 7, 3, 4, 3, 2, 1, 1, 0, 8, 8, 6, 6, 3, 2, 1, 1, 0, 9, 6, 9, 6, 5, 3, 2, 1, 1, 0, 10, 11, 10, 10, 8, 5, 3, 2, 1, 1, 0, 11, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 12, 17, 19, 19, 14, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Peter Luschny, Aug 24 2019

Dirichlet convolution of phi(n) and A008284(n,k) for n >= 1. - Richard L. Ollerton, May 07 2021

			Triangle starts:
[0] [1]
[1] [0, 1]
[2] [0, 2, 1]
[3] [0, 3, 1, 1]
[4] [0, 4, 3, 1, 1]
[5] [0, 5, 2, 2, 1, 1]
[6] [0, 6, 6, 4, 2, 1, 1]
[7] [0, 7, 3, 4, 3, 2, 1, 1]
[8] [0, 8, 8, 6, 6, 3, 2, 1, 1]
[9] [0, 9, 6, 9, 6, 5, 3, 2, 1, 1]

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A008284, A000010, A078392 (row sums), A282750.
Cf. A000041 (where reversed rows converge to).
T(2n,n) gives A052810.

def DivisorTriangle(f, T, Len, w = None):
    D = [[1]]
    for n in (1..Len-1):
        r = lambda k: [f(d)*T(n//d,k) for d in divisors(n)]
        L = [sum(r(k)) for k in (0..n)]
        if w != None: L = [*map(lambda v: v * w(n), L)]
        D.append(L)
    return D
DivisorTriangle(euler_phi, A008284, 10)

From Richard L. Ollerton, May 07 2021: (Start)
For n >= 1, T(n,k) = Sum_{i=1..n} A008284(gcd(n,i),k).
For n >= 1, T(n,k) = Sum_{i=1..n} A008284(n/gcd(n,i),k)*phi(gcd(n,i))/phi(n/gcd(n,i)). (End)

A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 5, 3, 1, 1, 0, 7, 3, 2, 1, 1, 0, 11, 6, 4, 2, 1, 1, 0, 15, 7, 5, 3, 2, 1, 1, 0, 22, 12, 7, 6, 3, 2, 1, 1, 0, 30, 14, 11, 7, 5, 3, 2, 1, 1, 0, 42, 22, 14, 11, 8, 5, 3, 2, 1, 1, 0, 56, 27, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 77, 40, 27, 21, 15, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Seiichi Manyama, May 14 2023

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  1,  1;
  0,  5,  3,  1, 1;
  0,  7,  3,  2, 1, 1;
  0, 11,  6,  4, 2, 1, 1;
  0, 15,  7,  5, 3, 2, 1, 1;
  0, 22, 12,  7, 6, 3, 2, 1, 1;
  0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Row sums give A323433.
Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.
T(2n,n) gives A052810.
Cf. A008284, A072233, A350890.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
    (j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, May 14 2023

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[j, b[n - j, Min[n - j, j]]][k*i], {i, 0, n/k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)

T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));

For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).

A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and partitions of n.

Original entry on oeis.org

4, 6, 8, 12, 16, 24, 32, 46, 62, 86, 114, 156, 204, 272, 354, 464, 596, 772, 982, 1256, 1586, 2006, 2512, 3152, 3918, 4874, 6022, 7438, 9132, 11210, 13686, 16700, 20288, 24622, 29768, 35956, 43276, 52032, 62372, 74678, 89168, 106350

Offset: 1

Omar E. Pol, Oct 29 2012

nonn

On the infinite square grid the diagram of regions of the set of partitions of n is represented by a rectangle with base = n and height = A000041(n). The rectangle contains n shells. Each shell contains regions. Each row of a region is a part. Each part of size k contains k cells. The number of regions equals the number of partitions of n (see illustrations in the links section). For a minimalist version see A139582. For the definition of "region of n" see A206437.

Omar E. Pol, Illustration of initial terms of A211026 and of A066186
Omar E. Pol, Illustration of the seven regions of 5

Cf. A000041, A052810, A135010, A139582, A141285, A186412, A186114, A187219, A193870, A194446, A194447, A206437, A211009

a(n) = 2*A000041(n) + 2 = 2*A052810(n) = A139582(n) + 2.

a(18) corrected by Georg Fischer, Apr 11 2024

A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -5, 10, -9, 3, 0, 1, -9, 36, -79, 98, -64, 17, 0, 1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0, 1, -28, 378, -3242, 19648, -88676, 306308, -819933, 1703404, -2723374, 3285552, -2887734, 1739326, -639065, 107435, 0

Offset: 1

Alois P. Heinz, Jun 15 2012

The ranked poset L(n) of partitions is defined in A002846. A partition of n into k parts is connected to another partition of n into k+1 parts that results from splitting one part of the first partition into two parts.

			L(5):     (32)---(221)
         /    \ /     \
        /      X       \
       /      / \       \
    (5)---(41)---(311)---(2111)---(11111)
Chromatic polynomial: q^7-9*q^6+36*q^5-79*q^4+98*q^3-64*q^2+17*q.
Triangle T(n,k) begins:
  1,   0;
  1,  -1,   0;
  1,  -2,   1,    0;
  1,  -5,  10,   -9,    3,     0;
  1,  -9,  36,  -79,   98,   -64,   17,     0;
  1, -17, 136, -666, 2192, -5032, 8111, -9013, 6569, -2818, 537, 0;

Alois P. Heinz, Rows n = 1..9, flattened
Olivier Gérard, The ranked posets L(2),...,L(8)
Eric Weisstein's World of Mathematics, Chromatic Polynomial
Wikipedia, Chromatic Polynomial

Row lengths give: 1+A000041(n) = A052810(n).
Row sums (for n>1) and last elements of rows give: A000004.
Columns k=1-2 give: A000012, (-1)*A000097(n-2).
Cf. A002846, A213242, A213385, A213427.

Edited by Alois P. Heinz at the suggestion of Gus Wiseman, May 02 2016

A281957 a(n) = largest k such that n has at least k partitions each containing at least k parts.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 12, 13, 14, 15, 16, 17, 18, 19, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61

Offset: 1

Arkadiusz Wesolowski, Feb 03 2017

			-------------------------------------
                          Number
Partitions of 5          of terms
-------------------------------------
5 .......................... 1
1 + 4 ...................... 2
2 + 3 ...................... 2
1 + 1 + 3 .................. 3
1 + 2 + 2 .................. 3
1 + 1 + 1 + 2 .............. 4
1 + 1 + 1 + 1 + 1 .......... 5
-------------------------------------
There are 7 partitions of the integer 5 is 7. The four partitions 1 + 1 + 3, 1 + 2 + 2, 1 + 1 + 1 + 2 and 1 + 1 + 1 + 1 + 1 each have at least 3 parts, so a(5) = 3.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Partition

Cf. A000070, A008284, A052810.

lst:=[]; k:=1; s:=0; for m in [0..8] do s+:=NumberOfPartitions(m); while k le s do Append(~lst, k); k+:=1; end while; Append(~lst, s); end for; lst;

A338001 Irregular triangle read by rows, a refinement of A271708.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 6, 2, 3, 0, 24, 8, 4, 3, 4, 0, 120, 8, 12, 6, 6, 4, 5, 0, 720, 48, 16, 48, 18, 6, 18, 8, 8, 5, 6, 0, 5040, 48, 48, 240, 18, 24, 12, 72, 12, 8, 24, 10, 10, 6, 7, 0, 40320, 384, 96, 192, 1440, 36, 36, 24, 36, 360, 32, 12, 32, 16, 96, 15, 10, 30, 12, 12, 7, 8

Offset: 0

Peter Luschny, Nov 13 2020

Row n of the triangle gives the sizes of the centralizers of any permutation of cycle type given by the partitions of n with max. part k.

T(n, k) divides n! if k > 0 and in this case the n!/T(n, k) give, up to order, the rows of A036039.

			Triangle rows start:
0: [1];
1: [0], [1];
2: [0], [2],    [2];
3: [0], [6],    [2],           [3];
4: [0], [24],   [8, 4],        [3],              [4];
5: [0], [120],  [8, 12],       [6, 6],           [4],         [5];
6: [0], [720],  [48, 16, 48],  [18, 6, 18],      [8, 8],      [5],      [6];
7: [0], [5040], [48, 48, 240], [18, 24, 12, 72], [12, 8, 24], [10, 10], [6], [7];
.
For n = 4 the partition of 4 with cycle type [2, 2] has centralizer size 8, and the partition [2, 1, 1] has centralizer size 4. Therefore in column 2 in the above triangle the pair [8, 4] appears.

S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.

Cf. A271708, A110143 (row sums), A052810 (row length), A126074, A036039.

def A338001(n):
    R = []
    for k in (0..n):
        P = Partitions(n, max_part=k, inner=[k])
        q = [p.aut() for p in P]
        R.append(q if q != [] else [0])
    return flatten(R)
for n in (0..7): print(A338001(n))

A356656 Partition triangle read by rows. The coefficients of the incomplete Bell polynomials.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 4, 3, 6, 1, 0, 1, 5, 10, 10, 15, 10, 1, 0, 1, 6, 15, 10, 15, 60, 15, 20, 45, 15, 1, 0, 1, 7, 21, 35, 21, 105, 70, 105, 35, 210, 105, 35, 105, 21, 1, 0, 1, 8, 28, 56, 35, 28, 168, 280, 210, 280, 56, 420, 280, 840, 105, 70, 560, 420, 56, 210, 28, 1

Offset: 0

Peter Luschny, Aug 28 2022

We call a triangle a 'partition triangle' if the rows have length A000041 or A000041 + 1.

			The triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1, 1;
[3] 0, 1, 3,  1;
[4] 0, 1, [4,  3],  6,  1;
[5] 0, 1, [5, 10], [10, 15],  10,  1;
[6] 0, 1, [6, 15, 10], [15,  60, 15], [20, 45],  15,   1;
[7] 0, 1, [7, 21, 35], [21, 105, 70, 105], [35, 210, 105], [35, 105], 21, 1;
Summing the bracketed terms reduces the triangle to A048993.
The first few polynomials are:
[0] 1;
[1] 0, z[0];
[2] 0, z[1], z[0]^2;
[3] 0, z[2], 3*z[0]*z[1], z[0]^3;
[4] 0, z[3], 4*z[0]*z[2]+3*z[1]^2, 6*z[0]^2*z[1], z[0]^4;
[5] 0, z[4], 5*z[0]*z[3]+10*z[1]*z[2], 10*z[0]^2*z[2]+15*z[0]*z[1]^2, 10*z[0]^3* z[1], z[0]^5;
It is noteworthy that the substitution z[n] -> n! for n >= 0 yields A132393. More examples are given in the authors blog post (see links).

E. T. Bell, Exponential polynomials, Ann. Math., 35 (1934), 258-277.
Peter Luschny, The Bell transform.

Variants: A036040, A080575, A178867. Row sums: A000110.

A048993 (reduced triangle), A052810 (length of rows), A132393 (factorial substituion).

aRow := n -> seq(coeffs(IncompleteBellB(n, k, seq(z[i], i = 0..n))), k = 0..n):
seq(aRow(n), n = 0..8);

from functools import cache
@cache
def incomplete_bell_polynomial(n, k):
    Z = var(["z_" + str(i) for i in range(n - k + 1)])
    R = PolynomialRing(ZZ, Z, n - k + 1, order='lex')
    if k == 0: return R(k^n)
    return R(sum(binomial(n-1,j-1) * incomplete_bell_polynomial(n-j,k-1) * Z[j-1]
            for j in range(n - k + 2)).expand())
def poly_row(n): return [incomplete_bell_polynomial(n, k) for k in range(n + 1)]
def coeff_row(n): return flatten([[0] if (c := p.coefficients()) == [] else c for p in poly_row(n)])
for n in range(8): print(coeff_row(n))

In row n the coefficients of IBell(n, k, Z_n) for k = 0..n are lined up. Z_n denotes the set of variables z[0], z[1], ... z[n] of the incomplete Bell polynomial IBell(n, k) of degree k.

cp's OEIS Frontend

A327029 T(n, k) = Sum_{d|n} phi(d) * A008284(n/d, k) for n >= 1, T(0, 0) = 1. Triangle read by rows for 0 <= k <= n.

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A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

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A211026 Number of segments needed to draw (on the infinite square grid) a diagram of regions and partitions of n.

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A213597 Triangle T(n,k), n>=1, 0<=k<=A000041(n), read by rows: row n gives the coefficients of the chromatic polynomial of the ranked poset L(n) of partitions of n, highest powers first.

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A281957 a(n) = largest k such that n has at least k partitions each containing at least k parts.

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Magma

A338001 Irregular triangle read by rows, a refinement of A271708.

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A356656 Partition triangle read by rows. The coefficients of the incomplete Bell polynomials.

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