|
| |
|
|
A003517
|
|
Number of permutations of [n+1] with exactly 1 increasing subsequence of length 3.
(Formerly M4177)
|
|
27
| |
|
|
1, 6, 27, 110, 429, 1638, 6188, 23256, 87210, 326876, 1225785, 4601610, 17298645, 65132550, 245642760, 927983760, 3511574910, 13309856820, 50528160150, 192113383644, 731508653106, 2789279908316, 10649977831752, 40715807302800
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 2,2
|
|
|
COMMENTS
| a(n-4) = number of n-th generation vertices in the tree of sequences with unit increase labeled by 5 (cf. Zoran Sunik reference) - Benoit Cloitre, Oct 07 2003
Number of standard tableaux of shape (n+3,n-2). - Emeric Deutsch, May 30 2004
|
|
|
REFERENCES
| 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) 743-751
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math. 152 (1996), no. 1-3, 307-313.
L. W. Shapiro, A Catalan triangle, Discrete Math. 14 (1976), no. 1, 83-90.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zoran Sunik, Self describing sequences and the Catalan family tree, Elect. J. Combin., 10 (No. 1, 2003).
|
|
|
LINKS
| D. Callan, A recursive bijective approach to counting permutations...
R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns
|
|
|
FORMULA
| a(n) = 6*C(2*n+1, n-2)/(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,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=5, a(n-3)=(-1)^(n-5)*coeff(charpoly(A,x),x^5). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
a(n)=sum(Catalan(i)*Catalan(j)*Catalan(k), i>=1,j>=1,k>=1, i+j+k=n+1)
|
|
|
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]] (* From Jean-François Alcover, Jul 13 2011, after g.f. *)
|
|
|
CROSSREFS
| T(n, n+6) for n=0, 1, 2, ..., array T as in A047072.
Cf. A001089, A084249.
First differences are in A026017.
A diagonal of any of the essentially equivalent arrays A009766, A030237, A033184, A059365, A099039, A106566, A130020, A047072.
Cf. A000108 A000245 A002057 A000344 A003517 A000588 A003518 A003519 A001392.
Sequence in context: A037604 A022634 A094788 * A108958 A005284 A198694
Adjacent sequences: A003514 A003515 A003516 * A003518 A003519 A003520
|
|
|
KEYWORD
| nonn,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
