

A003517


Number of permutations of [n+1] with exactly 1 increasing subsequence of length 3.
(Formerly M4177)


39



1, 6, 27, 110, 429, 1638, 6188, 23256, 87210, 326876, 1225785, 4601610, 17298645, 65132550, 245642760, 927983760, 3511574910, 13309856820, 50528160150, 192113383644, 731508653106, 2789279908316, 10649977831752, 40715807302800
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OFFSET

2,2


COMMENTS

a(n4) = number of nth generation vertices in the tree of sequences with unit increase labeled by 5 (cf. Zoran Sunic reference).  Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+3,n2).  Emeric Deutsch, May 30 2004
a(n) = A214292(2*n,n3) for n > 2.  Reinhard Zumkeller, Jul 12 2012
a(n) is the number of NorthEast paths from (0,0) to (n,n) that cross the diagonal y = x horizontally exactly once. By symmetry, it is also the number of NorthEast paths from (0,0) to (n,n) that cross the diagonal y = x vertically exactly once. Details can be found in Section 3.3 in Pan and Remmel's link.  Ran Pan, Feb 02 2016
a(n) is the number of permutations pi of [n+3] such that s(pi)=321456...(n+3), where s denotes West's stacksorting map.  Colin Defant, Jan 14 2019


REFERENCES

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


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Daniel Birmajer, Juan B. Gil, Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
D. Callan, A recursive bijective approach to counting permutations..., arXiv:math/0211380 [math.CO], 2002.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743751.
S. J. Cyvin, J. Brunvoll, E. Brendsdal, B. N. Cyvin and E. K. Lloyd, Enumeration of polyene hydrocarbons: a complete mathematical solution, J. Chem. Inf. Comput. Sci., 35 (1995) 743751. [Annotated scanned copy]
H. H. Gudmundsson, Dyck paths, standard Young tableaux, and pattern avoiding permutations, PU. M. A. Vol. 21 (2010), No. 2, pp. 265284 (see 4.3 p. 277).
R. K. Guy, Letter to N. J. A. Sloane, May 1990
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
V. E. Hoggatt, Jr., 7page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395405.
Markus Kuba, Alois Panholzer, Stirling permutations containing a single pattern of length three, Australasian Journal of Combinatorics, Vol. 74, No. 2 (2019), 216239.
N. Lygeros, O. Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math. 152 (1996), no. 13, 307313.
Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.
L. W. Shapiro, A Catalan triangle, Discrete Math., Vol. 14, No. 1 (1976), 8390.
L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 8390. [Annotated scanned copy]
Zoran Sunic, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (2003), N5.


FORMULA

a(n) = 6*binomial(2*n+1, n2)/(n+4).
G.f.: x^2*C(x)^6, where C(x) is g.f. for the Catalan numbers (A000108).  Emeric Deutsch, May 30 2004
E.g.f.: exp(2*x)*(Bessel_I(2,2*x)  Bessel_I(4,2*x)).  Paul Barry, Jun 04 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i1]=1, A[i,j]=Catalan(ji), (i<=j), and A[i,j]=0, otherwise. Then, for n >= 5, a(n3) = (1)^(n5)*coeff(charpoly(A,x),x^5).  Milan Janjic, Jul 08 2010
a(n) = Sum_{i>=1, j>=1, k>=1, i+j+k=n+1} Catalan(i)*Catalan(j)*Catalan(k).
(n+4)*(n2)*a(n) + 2*n*(2*n+1)*a(n1) = 0.  R. J. Mathar, Dec 04 2012


EXAMPLE

a(3)=6 because the only permutations of 1234 with exactly 1 increasing subsequence of length 3 are 1423, 4123, 1342, 2314, 2341, 3124.


MATHEMATICA

f[x_] = (Sqrt[1  4 x]  1)^6/(64 x^4); CoefficientList[Series[f[x], {x, 0, 25}], x][[3 ;; 26]] (* JeanFrançois Alcover, Jul 13 2011, after g.f. *)
Table[6 Binomial[2n+1, n2]/(n+4), {n, 2, 30}] (* Harvey P. Dale, Feb 27 2012 *)


PROG

(PARI) a(n)=6*binomial(2*n+1, n2)/(n+4) \\ Charles R Greathouse IV, May 18 2015
(PARI) x='x+O('x^50); Vec(x^2*((1(14*x)^(1/2))/(2*x))^6) \\ Altug Alkan, Nov 01 2015


CROSSREFS

T(n, n+6) for n=0, 1, 2, ..., array T as in A047072.
Cf. A001089, A084249, A000108, A000245, A002057, A000344, A000588, A003518, A003519, A001392.
First differences are in A026017.
A diagonal of any of the essentially equivalent arrays: A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Sequence in context: A094788 A221863 A216263 * A108958 A005284 A198694
Adjacent sequences: A003514 A003515 A003516 * A003518 A003519 A003520


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


STATUS

approved



