A014606 -id:A014606 - cp's OEIS frontend

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

1, 1, 0, 19, 1, 0, 1513, 166, 1, 0, 315523, 52715, 1361, 1, 0, 136085041, 30543236, 1528806, 10916, 1, 0, 105261234643, 29664031413, 2257312622, 42421946, 87375, 1, 0, 132705221399353, 45011574747714, 4637635381695, 153778143100, 1156669095, 699042, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

See the comments in A292604.

			Triangle starts:
[n\k][       0         1         2       3  4  5]
--------------------------------------------------
[0][         1]
[1][         1,        0]
[2][        19,        1,        0]
[3][      1513,      166,        1,     0]
[4][    315523,    52715,     1361,     1,  0]
[5][ 136085041, 30543236,  1528806, 10916,  1, 0]

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.
First column: A002115. Row sums: A014606. Alternating row sums: A292609.
Cf. A181985, A278073.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f),x):
A292605_row := proc(n) if n = 0 then return [1] fi;
add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292605_row(n) od;

# uses[A278073_row from A278073]
def A292605_row(n):
    if n == 0: return [1]
    L = A278073_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292605_row(n))

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) for n>0 and F_{3; 0}(x) = 1.

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))

A177291 Number of permutations of 3 copies of 1..n with all adjacent differences <= 1 in absolute value.

Original entry on oeis.org

1, 1, 20, 92, 506, 2288, 10010, 41618, 168284, 664958, 2584442, 9916688, 37679618, 142079906, 532572428, 1987037318, 7386724082, 27381500624, 101272019258, 373902595130, 1378571667644, 5077289249390, 18683930010890, 68709775705328, 252549056389394, 927895845621746

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (3n)!/6^n = A014606(n) for n<=2.

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=3 of A331562.

Cf. A014606.

a(0)=1 prepended by Alois P. Heinz, Jan 21 2020
a(19)-a(20) from Alois P. Heinz, Jan 22 2020
Terms a(21) and beyond from Andrew Howroyd, May 14 2020

A177605 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up, up.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 166923119, 134612399509, 176468898415460, 351456009607660440, 1010384984894493146400, 4028532188816344883922611, 21572165225621811583520157531, 151100918612288209338643024889160, 1354217120536797195675961944817053840

Offset: 0

R. H. Hardin, May 10 2010

nonn

Mingjia Yang and Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Cf. A014606.

Cf. A177596, A177615.

a(10) from Alois P. Heinz, Nov 01 2013

a(11)-a(13) from Alois P. Heinz, Aug 08 2018

A177615 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up, up, up.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137127810959, 182138398520387, 367988956112888200, 1074073326771101149080, 4350318942513425384777400, 23673963203663687592089088600, 168569155472096397352266896207243, 1536148007688582239667327040442799059

Offset: 0

R. H. Hardin, May 10 2010

nonn

Mingjia Yang and Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Cf. A014606.

Cf. A177596, A177605.

a(10)-a(13) from Alois P. Heinz, Aug 08 2018

A269113 Number of sequences with 3 copies each of 1,2,...,n avoiding the pattern 12...n.

Original entry on oeis.org

0, 0, 1, 374, 173891, 117392909, 117108036719, 171248808285596, 360953073372968159, 1072323973643442736211, 4376906243609822466600689, 23919710914189027455648239834, 170865299381465355439286870245691, 1561721420156259852074018974532765369

Offset: 0

Alois P. Heinz, Feb 19 2016

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..165

Column k=3 of A269129.

Cf. A014606, A047910.

a(n) = A014606(n) - A047910(n).

A327412 a(n) = multinomial(3*n+2; 2, 3, 3, ..., 3) (n times '3').

Original entry on oeis.org

1, 10, 560, 92400, 33633600, 22870848000, 26072766720000, 46174869861120000, 120054661638912000000, 438679733628584448000000, 2175851478797778862080000000, 14240947928731462652313600000000, 120136636726778618934917529600000000, 1280656547507460077846220865536000000000

Offset: 0

Peter Luschny, Sep 07 2019

nonn

Cf. A014606, A327411, A025035.

a:= n-> combinat[multinomial](3*n+2, 3$n, 2):
seq(a(n), n=0..17);  # Alois P. Heinz, Sep 07 2019

multinomial[n_, k_List] := n!/Times @@ (k!);
a[n_] := multinomial[3n+2, Join[{2}, Table[3, {n}]]];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Apr 19 2025 *)

def a(n): return multinomial([2] + [3] * n)
[a(n) for n in range(15)]

a(n) = 2^(-n-1)*3^(-n)*Gamma(3*n + 3).
a(n) = (9*(n-1)^3 + 36*(n-1)^2 + 47*n - 27)*a(n-1)/2 for n > 0.
a(n) / n! = A025035(n+1).
a(n)*(n+1) = A014606(n+1).

A357297 T(m,n) is the number of linear extensions of n fork-join DAGs of width m, read by downward antidiagonals.

Original entry on oeis.org

1, 1, 1, 6, 1, 1, 90, 20, 2, 1, 2520, 1680, 280, 6, 1, 113400, 369600, 277200, 9072, 24, 1, 7484400, 168168000, 1009008000, 163459296, 532224, 120, 1, 681080400, 137225088000, 9777287520000, 15205637551104, 237124952064, 49420800, 720, 1, 81729648000, 182509367040000, 207786914375040000, 4847253138540933120, 765985681152147456, 689598074880000, 6671808000, 5040, 1

Offset: 0

José E. Solsona, Feb 22 2023

The fork-join structure is a modeling structure, commonly seen for example in parallel computing, usually represented as a DAG (or poset). It has an initial "fork" vertex that spawns a number of m independent children vertices (the width) whose output edges are connected to a final "join" vertex. More generally, we can have a number n of these DAGs, each one with m+2 vertices.
The family of fork-join DAGs we are considering here can be depicted as follows (we omit the first column for n=0 because the graph is empty in this case):
m\n|    1    |      2        |          3
---------------------------------------------------
 0 |    o    |     o  o      |       o  o  o
   |    |    |     |  |      |       |  |  |
   |    o    |     o  o      |       o  o  o
---------------------------------------------------
   |    o    |     o  o      |       o  o  o
   |    |    |     |  |      |       |  |  |
 1 |    o    |     o  o      |       o  o  o
   |    |    |     |  |      |       |  |  |
   |    o    |     o  o      |       o  o  o
---------------------------------------------------
   |    o    |    o     o    |    o     o     o
   |   / \   |   / \   / \   |   / \   / \   / \
 2 |  o   o  |  o   o o   o  |  o   o o   o o   o
   |   \ /   |   \ /   \ /   |   \ /   \ /   \ /
   |    o    |    o     o    |    o     o     o
---------------------------------------------------
   |    o    |    o     o    |    o     o     o
   |   /|\   |   /|\   /|\   |   /|\   /|\   /|\
 3 |  o o o  |  o o o o o o  |  o o o o o o o o o
   |   \|/   |   \|/   \|/   |   \|/   \|/   \|/
   |    o    |    o     o    |    o     o     o
The array begins like this:
m\n|0   1         2               3                       4
-----------------------------------------------------------
 0 |1   1         6              90                    2520 ... A000680
 1 |1   1        20            1680                  369600 ... A014606
 2 |1   2       280          277200              1009008000 ... A260331
 3 |1   6      9072       163459296          15205637551104 ... A361901
 4 |1  24    532224    237124952064      765985681152147456 ... A362565
 5 |1 120  49420800 689598074880000 97981404549709824000000 ...
with columns: A000012 (n=0) and A000142 (n=1).

			T(3,1) = 6 is the number of linear extensions of one fork-join DAG of width 3. Let the DAG be labeled as follows:
     1
   / | \
  2  3  4
   \ | /
     5
Then the six linear extensions are:
  1 2 3 4 5
  1 2 4 3 5
  1 3 2 4 5
  1 3 4 1 5
  1 4 2 3 5
  1 4 3 2 5

Wikipedia, Fork-join model

Rows m = 0..4 give A000680, A014606, A260331, A361901, A362565.

Columns n = 0..1 give A000012, A000142.

(* Formula *)
T[m_, n_] := (n*(m+2))!/((m+1)^n*(m+2)^n)
(* 5 X 5 Table *)
Table[T[m, n], {m, 0, 5}, {n, 0, 5}]
(* Eight rows of the triangle *)
Table[Table[T[m, n - m], {m, 0, n}], {n, 0, 8}]
(* As a sequence *)
Flatten[Table[Table[T[m, n - m], {m, 0, n}], {n, 0, 8}]]

T(m,n) = (n*(m+2))!/((m+1)^n*(m+2)^n).

A375219 T(n,k) is the number of permutations of the multiset {1, 1, 1, 2, 2, 2, ..., n, n, n} with k occurrences of fixed triples (j,j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

Original entry on oeis.org

19, 1622, 57, 362997, 6488, 114, 166336604, 1814985, 16220, 190, 136221590695, 998019624, 5444955, 32440, 285, 181552310074386, 953551134865, 3493068684, 12704895, 56770, 399, 367942716863474473, 1452418480595088, 3814204539460, 9314849824, 25409790, 90832, 532

Offset: 2

Hugo Pfoertner, Aug 08 2024

Trivially, T(n,n) = 1 and T(n,n-1) = 0.

			The triangle begins
         19;
       1622,      57;
     362997,    6488,   114,
  166336604, 1814985, 16220, 190;
.
T(2,0) = 19: the permutations of {1,1,1,2,2,2} with no fixed triples are
[1,1,2,1,2,2], [1,1,2,2,1,2], [1,1,2,2,2,1], [1,2,1,1,2,2], [1,2,1,2,1,2], [1,2,1,2,2,1], [1,2,2,1,1,2], [1,2,2,1,2,1], [1,2,2,2,1,1], [2,1,1,1,2,2], [2,1,1,2,1,2], [2,1,1,2,2,1], [2,1,2,1,1,2], [2,1,2,1,2,1], [2,1,2,2,1,1], [2,2,1,1,1,2], [2,2,1,1,2,1], [2,2,1,2,1,1], [2,2,2,1,1,1].

Cf. A014606.

Cf. A374980, A375223 (columns 0 and 1 in a similar triangle for the multiset {1, 1, 2, 2, ..., n, n}).

mima (x, n1=1, i2=-oo) = {my (n2, n=#x, mi=x[n1], ma=mi); n2=if (i2<=0, n, min(n,i2)); for (i=n1+1, n2, if (x[i]ma, ma=x[i]))); [mi,ma]};
\\ returns row n of triangle, bsize is the block size in the multiset.
a375219(n, bsize=3) = {my (p=vector(bsize*n, i, 1+(i-1)\bsize), r=s=vector(n), m=vector(n-1)); forperm (p, q, for (b=1, n, my (bm=bsize*(b-1), j=mima(q, bm+1, bm+bsize)); r[b]=j[1]; s[b]=j[2]); my (rs=vector(n, i, r[i]==i && s[i]==i)); for (k=0 ,n-2, m[k+1]+=vecsum(rs)==k)); m}

Sum_{j=0..n-2} T(n,j) = (3*n)!/(6^n) - 1 = A014606(n) - 1.

More terms (three rows) from Alois P. Heinz, Aug 16 2024

A177295 Number of permutations of 3 copies of 1..n with all adjacent differences <= 5 in absolute value.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137225088000, 67280653440000, 22859094527068800, 6325233865588881600

Offset: 0

R. H. Hardin, May 06 2010

nonn

a(n) = (3n)!/6^n for n<=6.

Cf. A014606.

cp's OEIS Frontend

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

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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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A177291 Number of permutations of 3 copies of 1..n with all adjacent differences <= 1 in absolute value.

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Extensions

A177605 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up, up.

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Extensions

A177615 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up, up, up.

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Extensions

A269113 Number of sequences with 3 copies each of 1,2,...,n avoiding the pattern 12...n.

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A327412 a(n) = multinomial(3*n+2; 2, 3, 3, ..., 3) (n times '3').

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A357297 T(m,n) is the number of linear extensions of n fork-join DAGs of width m, read by downward antidiagonals.

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A375219 T(n,k) is the number of permutations of the multiset {1, 1, 1, 2, 2, 2, ..., n, n, n} with k occurrences of fixed triples (j,j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

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