

A033321


Binomial transform of Fine's sequence A000957: 1, 0, 1, 2, 6, 18, 57, 186, ...


22



1, 1, 2, 6, 21, 79, 311, 1265, 5275, 22431, 96900, 424068, 1876143, 8377299, 37704042, 170870106, 779058843, 3571051579, 16447100702, 76073821946, 353224531663, 1645807790529, 7692793487307, 36061795278341, 169498231169821
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OFFSET

0,3


COMMENTS

Number of permutations avoiding the patterns {2431,4231,4321}; number of weak sorting class based on 2431.  Len Smiley, Nov 01 2005
Number of permutations avoiding the patterns {2413, 3142, 2143}.  Vincent Vatter, Aug 16 2006
Number of permutations avoiding the patterns {2143, 3142, 4132}.  Alexander Burstein and Jonathan Bloom, Aug 03 2013
Number of unimodal Lehmer codes. Those are exactly the inversion sequences for permutations avoiding the patterns {2143, 3142, 4132}.  Alexander Burstein, Jun 16 2015
Number of skew Dyck paths of semilength n ending with a down step (1,1). A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the xaxis, consists of steps U=(1,1)(up), D=(1,1)(down) and L=(1,1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. Number of skew Dyck paths of semilength n and ending with a left step is A128714(n).  Emeric Deutsch, May 11 2007
Number of permutations sortable by a pop stack followed directly by a stack. Equivalently, the number of permutations avoiding {2431, 3142, 3241}.  Vincent Vatter, Mar 06 2013
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,...].  Philippe Deléham, Oct 24 2007
Starting with offset 1, Hankel transform = oddindexed Fibonacci numbers.  Gary W. Adamson, Dec 27 2008
Starting with offset 1 = INVERT transform of A002212: (1, 1, 3, 10, 36, 137, ...).  Gary W. Adamson, May 19 2009
Equals INVERTi transform of A007317: (1, 2, 5, 15, 51, 188, ...).  Gary W. Adamson, May 17 2009
Number of sequences (e(1), ..., e(n)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) > e(j) < e(k). [Martinez and Savage, 2.20]  Eric M. Schmidt, Jul 17 2017
From David Callan, Jul 21 2017: (Start)
a(n) is the number of permutations of [n] in which the excedances and subcedances are both increasing. (For example, the 3 permutations of [4] NOT counted by a(4)=21 are 3421, 4312, 4321 with excedances/subcedances 34/21, 43/12, 43/21 respectively.)
Proof. It suffices to show that (*) the number of such permutations of [n] containing k fixed points is binomial(n,k)*F(nk), where F is the Fine number A000957. Since F(n) is the number of 321avoiding derangements of [n] and because inserting or deleting a fixed point in a permutation does not change the excedance/fixed point/subcedance status of any other entry, (*) is an immediate consequence of the following claim: The excedances and subcedances of a permutation p are both increasing if and only if p avoids 321. The claim is a nice exercise utilizing the cycles of p for the "if" direction and the pigeonhole principle for the "only if" direction. (End)
Conjectured to be the number of permutations of length n that are sorted to the identity by a consecutive231avoiding stack followed by a classical21avoiding stack.  Colin Defant, Aug 30 2020
a(n) is the number of permutations of length n avoiding the partially ordered pattern (POP) {3>1, 3>4, 1>2} of length 4. That is, the number of length n permutations having no subsequences of length 4 in which the third element is the largest and the first element is larger than the second element.  Sergey Kitaev, Dec 10 2020


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0 to 200 by T. D. Noe)
M. Albert, R. Aldred, M. Atkinson, C Handley, D. Holton, D. McCaughan and H. van Ditmarsch, Sorting Classes, Elec. J. of Comb. 12 (2005) R31.
Christian Bean, Finding structure in permutation sets, Ph.D. Dissertation, Reykjavík University, School of Computer Science, 2018.
Christian Bean, Émile Nadeau, Henning Ulfarsson, Enumeration of Permutation Classes and Weighted Labelled Independent Sets, arXiv:1912.07503 [math.CO], 2019.
David Bevan, The permutation class Av(4213,2143), arXiv:1510.06328 [math.CO], 2014.
J. Bloom, A. Burstein, Egge triples and unbalanced Wilfequivalence, arXiv:1410.0230 [math.CO], 2014.
R. Brignall, S. Huczynska and V. Vatter, Simple permutations and algebraic generating functions, arXiv:math/0608391 [math.CO], 2006.
Robert Brignall, Jakub Sliacan, Juxtaposing Catalan permutation classes with monotone ones, arXiv:1611.05370 [math.CO], 2016.
C. Defant and K. Zheng, StackSorting with ConsecutivePatternAvoiding Stacks, arXiv:2008.12297 [math.CO], 2020.
Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 21912203.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, arXiv:1903.08946 [math.CO], 2019.
Alice L. L. Gao, Sergey Kitaev, On partially ordered patterns of length 4 and 5 in permutations, The Electronic Journal of Combinatorics 26(3) (2019), P3.26.
Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Toufik Mansour and Mark Shattuck, Nine classes of permutations enumerated by binomial transform of Fine's sequence, Discrete Applied Mathematics, Vol. 226, 31 July 2017, p. 94105.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Sam Miner, Enumeration of several twobyfour classes, arXiv preprint arXiv:1610.01908 [math.CO], 2016.
N. J. A. Sloane, Transforms
R. Smith and V. Vatter, A stack and a pop stack in series, arXiv:1303.1395 [math.CO], 2013.
Index entries for reversions of series


FORMULA

Also REVERT transform of x*(2*x1)/(x^2+x1).  Olivier Gérard
G.f.: 2/(1 + x + sqrt(1  6*x + 5*x^2)).
Dfinite with recurrence a(n) = ((13*n5)*a(n1)  (16*n23)*a(n2) + 5*(n2)*a(n3))/(2*(n+1)) (n>=3); a(0)=a(1)=1, a(2)=2.  Emeric Deutsch, Mar 21 2004
Binomial transform of Fine's sequence: a(n) = Sum_{k=0..n} binomial(n, k)*A000957(nk).
G.f.: 1/(1xx^2/(13xx^2/(13xx^2/(13xx^2/(1... (continued fraction).  Paul Barry, Jun 15 2009
a(n) = Sum_{k=0..n} A091965(n,k)*(2)^k.  Philippe Deléham, Nov 28 2009
a(n) = Sum_{m=1..n1} (Sum_(k=1..nm} (binomial(nm1, k1)*(m/(k+m))*binomial(2*k+m1, k+m1) ) ) + 1.  Vladimir Kruchinin, May 12 2011
a(n) = upper left term in M^n, M = the production matrix:
1, 1, 0, 0, 0, 0, 0, ...
1, 2, 1, 0, 0, 0, 0, ...
1, 2, 1, 1, 0, 0, 0, ...
1, 2, 1, 2, 1, 0, 0, ...
1, 2, 1, 2, 1, 1, 0, ...
1, 2, 1, 2, 1, 2, 1, ...
...
 Gary W. Adamson, Jul 08 2011
a(n) ~ 5^(n+3/2)/(18*sqrt(Pi)*n^(3/2)).  Vaclav Kotesovec, Aug 09 2013
G.f.: 1/(1x*C(x/(1x))), where C(x) = g.f. for A000108(n).  Alexander Burstein, Oct 05 2014


MAPLE

a[0] := 1: a[1] := 1: a[2] := 2: for n from 3 to 23 do a[n] := ((13*n5)*a[n1](16*n23)*a[n2]+5*(n2)*a[n3])/2/(n+1) od;


MATHEMATICA

f[n_] := Sum[Binomial[n, k]*g[n  k], {k, 0, n}]; g[n_] := Sum[(1)^(m + n)(n + m)!/n!/m!(n  m + 1)/(n + 1), {m, 0, n}]; Table[ f[n], {n, 24}] (* Robert G. Wilson v, Nov 04 2005 *)


PROG

(Maxima)
a(n):=sum(sum(binomial(nm1, k1)*m/(k+m)*binomial(2*k+m1, k+m1), k, 1, nm), m, 1, n1)+1; /* Vladimir Kruchinin, May 12 2011 */
(PARI) a(n)=1+sum(m=1, n1, sum(k=1, nm, binomial(nm1, k1)/(k+m)* binomial(2*k+m1, k+m1)*m)) \\ Charles R Greathouse IV, Mar 06 2013
(PARI) x='x+O('x^50); Vec(2/(1+x+sqrt(16*x+5*x^2))) \\ Altug Alkan, Oct 22 2015


CROSSREFS

Cf. A000957, A002212, A007317, A128714, A214611.
Sequence in context: A150197 A150198 A257562 * A050203 A112806 A150199
Adjacent sequences: A033318 A033319 A033320 * A033322 A033323 A033324


KEYWORD

nonn


AUTHOR

Emeric Deutsch


EXTENSIONS

More terms from Robert G. Wilson v, Nov 04 2005
Entry revised by N. J. A. Sloane, Aug 07 2006


STATUS

approved



