A178803 -id:A178803 - cp's OEIS frontend

A327022 Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.

1, 1, 1, 6, 1, 30, 90, 1, 56, 70, 1260, 2520, 1, 90, 420, 3780, 9450, 75600, 113400, 1, 132, 990, 924, 8910, 83160, 34650, 332640, 1247400, 6237000, 7484400, 1, 182, 2002, 6006, 18018, 270270, 252252, 630630, 1081080, 15135120, 12612600, 37837800, 189189000, 681080400, 681080400

Offset: 0

Peter Luschny, Aug 27 2019

We call an irregular triangle T a partition triangle if T(n, k) is defined for n >= 0 and 0 <= k < A000041(n).

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 2. For instance 2*P(4, .) = [[8], [6, 2], [4, 4], [4, 2, 2], [2, 2, 2, 2]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;   6]
[3] [1;  30;  90]
[4] [1;  56,  70; 1260; 2520]
[5] [1;  90, 420; 3780, 9450; 75600; 113400]
[6] [1; 132, 990,  924; 8910, 83160,  34650; 332640, 1247400; 6237000; 7484400]
.
T(4, 1) = 56 because [6, 2] is the integer partition 2*P(4, 1) in the canonical order and there are 28 set partitions which have the shape [6, 2] (an example is {{1, 3, 4, 5, 6, 8}, {2, 7}}). Finally, since the order of the sets is taken into account, one gets 2!*28 = 56.

Row sums: A094088, alternating row sums: A028296, main diagonal: A000680, central column A281478, by length: A241171.
Cf. A178803 (m=0), A133314 (m=1), this sequence (m=2), A327023 (m=3), A327024 (m=4).
Cf. A080577, A000041, A072233.

def GenOrdSetPart(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return [factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes]
def A327022row(n): return GenOrdSetPart(2, n)
for n in (0..6): print(A327022row(n))

A318144 T(n, k) = (-1)^kk!P(n, k), where P(n, k) is the number of partitions of n of length k. Triangle read by rows, 0 <= k <= n.

Original entry on oeis.org

1, 0, -1, 0, -1, 2, 0, -1, 2, -6, 0, -1, 4, -6, 24, 0, -1, 4, -12, 24, -120, 0, -1, 6, -18, 48, -120, 720, 0, -1, 6, -24, 72, -240, 720, -5040, 0, -1, 8, -30, 120, -360, 1440, -5040, 40320, 0, -1, 8, -42, 144, -600, 2160, -10080, 40320, -362880

Offset: 0

Peter Luschny, Aug 20 2018

			[0] [1],
[1] [0, -1],
[2] [0, -1, 2],
[3] [0, -1, 2,  -6],
[4] [0, -1, 4,  -6,  24],
[5] [0, -1, 4, -12,  24, -120],
[6] [0, -1, 6, -18,  48, -120,  720],
[7] [0, -1, 6, -24,  72, -240,  720,  -5040],
[8] [0, -1, 8, -30, 120, -360, 1440,  -5040, 40320],
[9] [0, -1, 8, -42, 144, -600, 2160, -10080, 40320, -362880]

Alois P. Heinz, Rows n = 0..150, flattened (first 45 rows from Peter Luschny)

Row sums are A260845, absolute row sums are A101880.

Cf. A008284, A072233, A178803.

/* As triangle: */
[[(-1)^k*#Partitions(n,k)*Factorial(k): k in [0..n]]: n in [0..10]]; // Bruno Berselli, Aug 20 2018

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>1,
      b(n, i-1), 0)+expand(b(n-i, min(n-i, i))*x))
    end:
T:= n-> (p-> seq(i!*coeff(p, x, i)*(-1)^i, i=0..n))(b(n$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Sep 18 2019

t[n_, k_] := (-1)^k  k! (IntegerPartitions[n, {k}] // Length);
Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > 1,
     b[n, i - 1], 0] + Expand[b[n - i, Min[n - i, i]]*x]];
T[n_] := Function[p, Table[i!*Coefficient[p, x, i]*(-1)^i, {i, 0, n}]][ b[n, n]];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

from sage.combinat.partition import number_of_partitions_length
def A318144row(n):
    return [(-1)^k*number_of_partitions_length(n, k)*factorial(k) for k in (0..n)]
for n in (0..9): print(A318144row(n))

A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 20, 1, 168, 1680, 1, 440, 924, 55440, 369600, 1, 910, 10010, 300300, 1261260, 33633600, 168168000, 1, 1632, 37128, 48620, 1113840, 24504480, 17153136, 326726400, 2058376320, 34306272000, 137225088000

Offset: 0

Peter Luschny, Aug 27 2019

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 3. For instance 3*P(4, .) = [[12], [9, 3], [6, 6], [6, 3, 3], [3, 3, 3, 3]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;   20]
[3] [1;  168;  1680]
[4] [1;  440,   924;  55440;  369600]
[5] [1;  910, 10010; 300300, 1261260; 33633600; 168168000]
[6] [1; 1632, 37128,  48620; 1113840, 24504480,  17153136; 326726400, 2058376320;
     34306272000; 137225088000]
.
T(4, 1) = 440 because [9, 3] is the integer partition 3*P(4, 1) in the canonical order and there are 220 set partitions which have the shape [9, 3]. Finally, since the order of the sets is taken into account, one gets 2!*220 = 440.

Row sums: A243664, alternating row sums: A002115, main diagonal: A014606, central column A281479, by length: A278073.
Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), this sequence (m=3), A327024 (m=4).
Cf. A080577, A000041, A072233.

# uses[GenOrdSetPart from A327022]
def A327023row(n): return GenOrdSetPart(3, n)
for n in (0..6): print(A327023row(n))

A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 70, 1, 990, 34650, 1, 3640, 12870, 2702700, 63063000, 1, 9690, 251940, 26453700, 187065450, 17459442000, 305540235000, 1, 21252, 1470942, 2704156, 154448910, 8031343320, 9465511770, 374796021600, 3975514943400, 231905038365000, 3246670537110000

Offset: 0

Peter Luschny, Aug 27 2019

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 4. For instance 4*P(4, .) = [[16], [12, 4], [8, 8], [8, 4, 4], [4, 4, 4, 4]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;    70]
[3] [1;   990;   34650]
[4] [1;  3640,   12870;  2702700;  63063000]
[5] [1;  9690,  251940; 26453700, 187065450; 17459442000; 305540235000]
[6] [1; 21252, 1470942,  2704156; 154448910,  8031343320,   9465511770;
     374796021600, 3975514943400; 231905038365000; 3246670537110000]
.
T(4, 1) = 3640 because [12, 4] is the integer partition 4*P(4, 1) in the canonical order and there are 1820 set partitions which have the shape [12, 4]. Finally, since the order of the sets is taken into account, one gets 2!*1820 = 3640.

Row sums: A243665, alternating row sums: A211212, main diagonal: A014608, central column: A281480, by length: A278074.
Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), A327023 (m=3), this sequence (m=4).
Cf. A080577, A000041, A072233.

# uses[GenOrdSetPart from A327022]
def A327024row(n): return GenOrdSetPart(4, n)
for n in (0..6): print(A327024row(n))

A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

Original entry on oeis.org

1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, 2, 3, 1, -1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, -2, -1, -2, -1, 0, 6, 3, 8, 6, 1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 2, 1, 2, 1, 0, 0, 0, 2, 2, 1, 0, 1, 0, 0, -6, -6, -5, -3, -3, -1, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):  -1
  (11):  1  1
.
  (3):    1
  (21):  -1 -1
  (111):  2  3  1
.
  (4):    -1
  (22):    1  1
  (31):    1     1
  (211):  -2 -1 -2 -1
  (1111):  6  3  8  6  1
.
  (5):      1
  (41):    -1 -1
  (32):    -1    -1
  (221):    2  1  2  1
  (311):    2  2  1     1
  (2111):  -6 -6 -5 -3 -3 -1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: F(32) = -p(5) - p(32).

Wikipedia, Symmetric polynomial

Row sums are A178803. Up to sign, same as A321931. This is a regrouping of the triangle A321899.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

cp's OEIS Frontend

A327022 Partition triangle read by rows. Number of ordered set partitions of the set {1, 2, ..., 2*n} with all block sizes divisible by 2.

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Sage

A318144 T(n, k) = (-1)^k*k!*P(n, k), where P(n, k) is the number of partitions of n of length k. Triangle read by rows, 0 <= k <= n.

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A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

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A327024 Ordered set partitions of the set {1, 2, ..., 4*n} with all block sizes divisible by 4, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

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Sage

A321934 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in F(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and F is augmented forgotten symmetric functions.

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A318144 T(n, k) = (-1)^kk!P(n, k), where P(n, k) is the number of partitions of n of length k. Triangle read by rows, 0 <= k <= n.