OFFSET
0,3
REFERENCES
Miklós Bóna, Combinatorics of Permutations. Discrete Mathematics and its Applications (Boca Raton), 2nd edn. CRC Press, Boca Raton (2012).
LINKS
David Bevan, Table of n, a(n) for n = 0..50 (from the Conway/Guttmann reference; terms 0..31 by Joerg Arndt, taken from the Johansson/Nakamura reference; terms 37..50 by Bjarki Ágúst Guðmundsson, taken from the Conway/Guttmann/Zinn-Justin reference).
M. H. Albert, M. Elder, A. Rechnitzer, P. Westcott, and M. Zabrocki, On the Wilf-Stanley limit of 4231-avoiding permutations and a conjecture of Arratia, arXiv:math/0502504 [math.CO], 2005.
R. Arratia, On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern., Electron. J. Combin. 6, N1 (1999).
David Bevan, Permutations avoiding 1324 and patterns in Łukasiewicz paths, arXiv:1406.2890 [math.CO], 2014-2015.
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, European J. Combin. 88 (2020).
Miklós Bóna, A new upper bound for 1324-avoiding permutations, arXiv:1207.2379 [math.CO], 2012.
Miklós Bóna, A new upper bound for 1324-avoiding permutations, Combin. Probab. Comput. 23(5), 717-724 (2014).
Miklós Bóna, A new record for 1324-avoiding permutations, arXiv:1404.4033 [math.CO], 2014.
Miklós Bóna, A new record for 1324-avoiding permutations, European Journal of Mathematics (2015) 1:198-206, DOI 10.1007/s40879-014-0020-6.
A. Claesson, V. Jelínek, and E. Steingrímsson, Upper bounds for the Stanley-Wilf limit of 1324 and other layered patterns. J. Combin. Theory Ser. A 119(8), 1680-1691 (2012).
Andrew R. Conway and Anthony J. Guttmann, On the growth rate of 1324-avoiding permutations, arXiv:1405.6802 [math.CO], (2014).
Andrew R. Conway, Anthony J. Guttmann and Paul Zinn-Justin, 1324-avoiding permutations revisited, arXiv preprint arXiv:1709.01248 [math.CO], 2017.
Steven Finch, Pattern-Avoiding Permutations [Broken link?]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
A. L. L. Gao, S. Kitaev, and P. B. Zhang, On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
Ruth Hoffmann, Özgür Akgün, and Christopher Jefferson, Composable Constraint Models for Permutation Enumeration, arXiv:2311.17581 [cs.DM], 2023.
Fredrik Johansson and Brian Nakamura, Using functional equations to enumerate 1324-avoiding permutations, arXiv:1309.7117 [math.CO], (2013).
A. Marcus and G. Tardos, Excluded permutation matrices and the Stanley-Wilf conjecture, J. Combin. Theory Ser. A 107(1), 153-160 (2004).
B. K. Nakamura, Computational methods in permutation patterns, PhD Dissertation, Rutgers University, May 2013.
Brian Nakamura and Doron Zeilberger, Using Noonan-Zeilberger Functional Equations to enumerate (in Polynomial Time!) Generalized Wilf classes, arXiv preprint arXiv:1209.2353 [math.CO], 2012.
Anthony Zaleski and Doron Zeilberger, On the Intriguing Problem of Counting (n+1,n+2)-Core Partitions into Odd Parts, arXiv:1712.10072 [math.CO], 2017.
EXAMPLE
a(4)=23 because all 24 permutations of length 4, except 1324 itself, avoid the pattern 1324.
MAPLE
count1324 := proc(n::nonnegint) if (n<4) then return n!; fi; if (n=4) then return 23; fi; return nodes([5, 5, 5, 5], n-5) + nodes([5, 3, 5, 5], n-5) + nodes([5, 4, 4, 5], n-5) + nodes([5, 5, 4, 5], n-5) + nodes([4, 3, 4], n-5) + nodes([5, 3, 4, 5], n-5); end:
nodes := proc(p, h) option remember; local i, j, s, l; if (h=0) then return convert(p, `+`); fi; s := 0; for j to nops(p) do l := p[j]+1; for i from 2 to j do l := l, `min`(j+1, p[i]); od; for i from j+1 to p[j] do l := l, p[i-1]+1; od; s := s+nodes([l], h-1); od; return s; end:
MATHEMATICA
a[n_] := n!/; n<4; a[4]=23; a[n_] := Total[nodes[#, n-5]&/@{{4, 3, 4}, {5, 3, 4, 5}, {5, 3, 5, 5}, {5, 4, 4, 5}, {5, 5, 4, 5}, {5, 5, 5, 5}}]; nodes[p_, 0]:=Total[p]; nodes[p_, h_] := nodes[p, h] = Sum[nodes[Join[{p[[j]]+1}, Min[j+1, #]&/@p[[2;; j]], p[[j;; p[[j]]-1]]+1], h-1], {j, Length[p]}]; Array[a, 12] (* David Bevan, May 25 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Darko Marinov (marinov(AT)lcs.mit.edu), May 17 2001
EXTENSIONS
More terms from Vincent Vatter, Feb 26 2005
a(23)-a(25) added from the Albert et al. paper by N. J. A. Sloane, Mar 29 2013
STATUS
approved