A092920 -id:A092920 - cp's OEIS frontend

A007476 Shifts 2 places left under binomial transform.

1, 1, 1, 2, 4, 9, 23, 65, 199, 654, 2296, 8569, 33825, 140581, 612933, 2795182, 13298464, 65852873, 338694479, 1805812309, 9963840219, 56807228074, 334192384460, 2026044619017, 12642938684817, 81118550133657, 534598577947465, 3615474317688778, 25070063421597484

Offset: 0

N. J. A. Sloane

Starting (1, 2, 4, 9, 23, ...) = row sums of triangle A153859. - Gary W. Adamson, Jan 02 2009
Binomial transform of the sequence starting (1, 1, 2, 4, 9, ...) = first differences of (1, 2, 4, 9, 23, ...); that is, (1, 2, 5, 14, 42, 134, 455, 1642, ...). - Gary W. Adamson, May 20 2013
Row sums of triangle A256161. - Margaret A. Readdy, Mar 16 2015
RG-words corresponding to set partitions of {1, ..., n} with every even entry appearing exactly once. - Margaret A. Readdy, Mar 16 2015
a(n) is the number of partitions of [n] whose blocks can be written such that the smallest elements form an increasing sequence and the largest elements form a decreasing sequence. a(5) = 9: 12345, 1235|4, 1245|3, 125|34, 1345|2, 135|24, 145|23, 15|234, 15|24|3. - Alois P. Heinz, Apr 24 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..650 (first 101 terms from T. D. Noe)
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210.
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
Yue Cai and Margaret Readdy, Negative q-Stirling numbers, arXiv:1506.03249 [math.CO], 2015.
A. Claesson and T. Mansour, Permutations avoiding a pair of Babson-Steingrimsson patterns, arXiv:math/0107044 [math.CO], 2001-2010.
Rigoberto Flórez, José L. Ramírez, Fabio A. Velandia, and Diego Villamizar, Some Connections Between Restricted Dyck Paths, Polyominoes, and Non-Crossing Partitions, arXiv:2308.02059 [math.CO], 2023. See Table 1 p. 13.
N. J. A. Sloane, Transforms
L. Sze, OEIS conjecture 70

Cf. A000994, A000995, A092920, A153859.

Row sums of A246118.

a:= proc(n) option remember; `if`(n<2, 1,
      add(a(j)*binomial(n-2, j), j=0..n-2))
    end:
seq(a(n), n=0..31);  # Alois P. Heinz, Jul 29 2019

a[0] = a[1] = 1; a[n_] := a[n] = Sum[Binomial[n-2, k] a[k], {k, 0, n-2}]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 08 2012, after Ralf Stephan *)

a(n)=if(n<2, 1, sum(k=0, n-2, binomial(n-2, k)*a(k))) /* Ralf Stephan; corrected by Manuel Blum, May 22 2010 */

G.f.: Sum_{k>=0} x^(2k)/(Product_{m=0..k-1} (1-mx) * Product_{m=0..k+1} (1-mx)).
G.f. A(x) satisfies A(x) = 1 + x + (x^2/(1-x))*A(x/(1-x)). - Vladimir Kruchinin, Nov 28 2011
a(n) = A000994(n) + A000995(n). - Peter Bala, Jan 27 2015

Spelling correction by Jason G. Wurtzel, Aug 22 2010

A362549 Number of partitions of [n] whose blocks can be ordered such that the i-th block (except possibly the last) has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 64, 187, 566, 1777, 5820, 19944, 71343, 264719, 1011292, 3953381, 15756609, 63945484, 264384828, 1115246518, 4806957739, 21189601861, 95516470253, 439777682222, 2064164172616, 9853934668051, 47736608806520, 234235866539512, 1162618720397931

Offset: 0

Alois P. Heinz, Apr 24 2023

nonn

			a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 2: 12, 1|2.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 9: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 1|23|4.
a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45.
a(6) = 64: 123456, 12345|6, 12346|5, 1234|56, 12356|4, ..., 1|2356|4, 1|235|46, 16|23|45, 1|236|45, 1|23|456.

Alois P. Heinz, Table of n, a(n) for n = 0..888
Wikipedia, Partition of a set

Cf. A000110, A007476, A092920, A210540, A362635, A362637, A362893.

b:= proc(n, t) option remember; `if`(n<=t, 1,
      add(b(j, t+1)*binomial(n-t, j), j=0..n-t))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);

A346660 Number of cyclic patterns of length n that avoid the vincular pattern 23-1-4.

Original entry on oeis.org

1, 1, 1, 2, 5, 14, 42, 133, 442, 1537, 5583, 21165, 83707, 345324, 1485687, 6663354, 31134078, 151408319, 765462514, 4017644518, 21860398111, 123120413119, 716701884408, 4305828784896, 26661920519485, 169937265101628, 1113616036893636, 7494786443901137

Offset: 0

Rupert Li, Aug 03 2021

nonn

The vincular pattern 23-1-4 requires the 2 and the 3 to be adjacent.

By the trivial Wilf equivalence obtained by reversing the permutations, a(n) is also the number of cyclic patterns of length n that avoid the vincular pattern 32-4-1.

Vaclav Kotesovec, Table of n, a(n) for n = 0..500
Rupert Li, Vincular Pattern Avoidance on Cyclic Permutations, arXiv:2107.12353 [math.CO], 2021.

Cf. A025242, A047970, A092920, A346661.

For n >= 2, a(n) = Sum_{i=0..n-2} binomial(n-2,i) * A092920(i).

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A007476 Shifts 2 places left under binomial transform.

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A362549 Number of partitions of [n] whose blocks can be ordered such that the i-th block (except possibly the last) has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.

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Maple

A346660 Number of cyclic patterns of length n that avoid the vincular pattern 23-1-4.

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Author

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