A014606 -id:A014606 - cp's OEIS frontend

A177607 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, up.

Original entry on oeis.org

1, 1, 20, 1680, 359582, 156245130, 120386806922, 150097348110878, 283411248751268531

Offset: 0

R. H. Hardin, May 10 2010

nonn

Cf. A014606.

A177612 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, up, up, up.

Original entry on oeis.org

1, 1, 20, 1680, 362082, 159459279, 124951069219, 158754672447498, 305860462403616044

Offset: 0

R. H. Hardin, May 10 2010

nonn

Cf. A014606.

A177616 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, down, down.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137150254800, 182209829636934, 368221858096998960

Offset: 0

R. H. Hardin, May 10 2010

nonn

Cf. A014606.

A177617 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, down, up.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 167510862, 135702841152, 178756061502340, 357729925926146006

Offset: 0

R. H. Hardin, May 10 2010

nonn

Cf. A014606.

A177629 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, up, up, up, up.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 167714730, 136194434423, 179979692293317, 361532596882040400

Offset: 0

R. H. Hardin, May 10 2010

nonn

Cf. A014606.

A177630 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, down, down, down, down.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 167895912, 136567215720, 180834782762247, 364064983677371040

Offset: 0

R. H. Hardin, May 10 2010

Cf. A014606.

A177635 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 168168000, 137225088000, 182499151015439, 369333660414653745, 1080107104118231632500, 4384231121059173932562000, 23913914175434871142808715000, 170693577054027116430454774306800, 1559452501977701854639515593122328400

Offset: 0

R. H. Hardin, May 10 2010

nonn

Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.

Cf. A014606.

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

A211310 a(n) = number |fdw(P,(n))| of entangled P-words with s=3.

Original entry on oeis.org

1, 18, 1566, 354456, 163932120, 134973740880, 180430456454640, 366311352681348480

Offset: 1

N. J. A. Sloane, Apr 08 2012

nonn

See Jenca and Sarkoci for the precise definition.

Gejza Jenca and Peter Sarkoci, Linear extensions and order-preserving poset partitions, arXiv preprint arXiv:1112.5782, 2011

Cf. A014606, A025035.

From Peter Bala, Sep 05 2012: (Start)
Conjectural e.g.f.: 2 - 1/A(x), where A(x) = sum {n = 0..inf} (3*n)!/6^n*x^n/n! is the e.g.f. for A014606 (also the o.g.f. for A025035).
If true, this leads to the recurrence equation: a(n) = (3*n)!/6^n - sum {k = 1..n-1} (3*k)!/6^k*binomial(n,k)*a(n-k) with a(1) = 1.
(End)

A266738 Number of words on {1,1,1,2,2,2,3,3,3,4,4,4,...,n,n,n} avoiding the pattern 12345.

Original entry on oeis.org

1, 1, 20, 1680, 369600, 117392909, 46121962742, 21198300356500, 11003612776114008, 6290031043253973544, 3887357166155963541538, 2562077915376091538040250, 1782153151031487742187453640, 1297781266782084301101836538690, 983066960483171632842827775906144

Offset: 0

N. J. A. Sloane, Jan 06 2016

nonn

Ferenc Balogh, A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length, preprint arXiv:1505.01389, 2015.
Shalosh B. Ekhad and Doron Zeilberger, The Generating Functions Enumerating 12..d-Avoiding Words with r occurrences of each of 1,2, ..., n are D-finite for all d and all r, 2014

Cf. A014606, A220097, A266734, A266735, A266736-A266741.

More terms from Alois P. Heinz, Jan 14 2016

A327410 Numbers represented by the partition coefficients of prime partitions.

Original entry on oeis.org

1, 6, 10, 20, 21, 36, 56, 78, 90, 105, 120, 171, 210, 252, 300, 364, 465, 528, 560, 741, 756, 792, 903, 990, 1140, 1176, 1485, 1540, 1680, 1830, 1953, 1980, 2346, 2520, 2600, 2628, 2775, 3240, 3432, 3570, 4095, 4368, 4851, 4960, 5253, 5460, 5886, 5984, 6105

Offset: 1

Peter Luschny, Sep 07 2019

nonn

Given a partition pi = (p1, p2, p3, ...) we call the associated multinomial coefficient (p1+p2+ ...)! / (p1!*p2!*p3! ...) the 'partition coefficient' of pi and denote it by . We say 'k is represented by pi' if k = .

A partition is a prime partition if all parts are prime.

			(2*n)!/2^n (for n >= 1) is a subsequence because [2,2,...,2] (n times '2') is a prime partition. Similarly A327411(n) is a subsequence because [3,2,2,...,2] (n times '2') is a prime partition. (3*n)!/(6^n) and A327412 are subsequences for the same reason.
The representations are not unique. 1 is the represented by all partitions of the form [p], p prime. For example 210 is represented by [3, 2, 2] and by [19, 2]. The list below shows the partitions with the smallest sum.
1   <- [2],
6   <- [2, 2],
10  <- [3, 2],
20  <- [3, 3],
21  <- [5, 2],
36  <- [7, 2],
56  <- [5, 3],
78  <- [11, 2],
90  <- [2, 2, 2],
105 <- [13, 2],
120 <- [7, 3],
171 <- [17, 2],
210 <- [3, 2, 2],
252 <- [5, 5],
300 <- [23, 2].

George E. Andrews, Arnold Knopfmacher, and Burkhard Zimmermann, On the number of distinct multinomial coefficients, arXiv:math/0509470 [math.CO], 2005.
Eric Weisstein's World of Mathematics, Prime Partition

Cf. A000607, A036038, A325306, A000680, A327411, A014606, A327412.

def A327410_list(n):
    res = []
    for k in range(2*n):
        P = Partitions(k, parts_in = prime_range(k+1))
        res += [multinomial(p) for p in P]
    return sorted(Set(res))[:n]
print(A327410_list(20))

cp's OEIS Frontend

A177607 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, up.

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A177612 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, up, up, up.

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A177616 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, down, down.

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A177617 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, down, down, down, up.

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A177629 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, down, up, up, up, up.

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A177630 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, down, down, down, down.

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A177635 Number of permutations of 3 copies of 1..n avoiding adjacent step pattern up, up, up, up, up, up.

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Extensions

A211310 a(n) = number |fdw(P,(n))| of entangled P-words with s=3.

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Formula

A266738 Number of words on {1,1,1,2,2,2,3,3,3,4,4,4,...,n,n,n} avoiding the pattern 12345.

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Extensions

A327410 Numbers represented by the partition coefficients of prime partitions.

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SageMath