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A003319
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Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.
(Formerly M2948)
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110
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1, 1, 1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, 5201061455, 73943424413, 1123596277863, 18176728317413, 311951144828863, 5661698774848621, 108355864447215063, 2181096921557783605, 46066653228356851631, 1018705098450570562877
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OFFSET
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0,4
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COMMENTS
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Also the number of permutations with no global descents, as introduced by Aguiar and Sottile [Corollaries 6.3, 6.4 and Remark 6.5].
Also the dimensions of the homogeneous components of the space of primitive elements of the Malvenuto-Reutenauer Hopf algebra of permutations. This result, due to Poirier and Reutenauer [Theoreme 2.1] is stated in this form in the work of Aguiar and Sottile [Corollary 6.3] and also in the work of Duchamp, Hivert and Thibon [Section 3.3].
Related to number of subgroups of index n-1 in free group of rank 2 (i.e., maximal number of subgroups of index n-1 in any 2-generator group). See Problem 5.13(b) in Stanley's Enumerative Combinatorics, Vol. 2.
For every n, a(n+1) is also the moment of order n for the probability density function rho(x) = exp(x)/(Ei(1,-x)*(Ei(1,-x) + 2*I*Pi)) on the interval 0..infinity, with Ei the exponential-integral function. - Groux Roland, Jan 16 2009
Also (apparently), a(n+1) is the number of rooted hypermaps with n darts on a surface of any genus (see Walsh 2012). - N. J. A. Sloane, Aug 01 2012
Also recurrent sequence A233824 (for n > 0) in Panaitopol's formula for pi(x), the number of primes <= x. - Jonathan Sondow, Dec 19 2013
Also the number of mobiles (cyclic rooted trees) with an arrow from each internal vertex to a descendant of that vertex. - Brad R. Jones, Sep 12 2014
Up to sign, Möbius numbers of the shard intersection orders of type A, see Theorem 1.3 in Reading reference. - F. Chapoton, Apr 29 2015
Also, a(n) is the number of distinct leaf matrices of complete non-ambiguous trees of size n. - Daniel Chen, Oct 23 2022
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REFERENCES
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Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 25, Example 20.
E. W. Bowen, Letter to N. J. A. Sloane, Aug 27 1976.
L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 84 (#25), 262 (#14) and 295 (#16).
P. de la Harpe, Topics in Geometric Group Theory, Univ. Chicago Press, 2000, p. 23, N_{n,2}.
I. M. Gessel and R. P. Stanley, Algebraic Enumeration, chapter 21 in Handbook of Combinatorics, Vol. 2, edited by R. L. Graham et al., The MIT Press, Mass, 1995.
M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 22.
H. P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 1, Chap. 1, Ex. 128; Vol. 2, 1999, see Problem 5.13(b).
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LINKS
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Marcelo Aguiar and Aaron Lauve, Convolution Powers of the Identity, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. Discrete Mathematics and Theoretical Computer Science, pp.1053-1064, 2013, DMTCS Proceedings. <hal-01229682>.
L. Comtet, Series inversions, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)
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FORMULA
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G.f.: 2 - 1/Sum_{k>=0} k!*x^k.
Also a(n) = n! - Sum_{k=1..n-1} k!*a(n-k) [Bowen, 1976].
Also coefficients in the divergent series expansion log Sum_{n>=0} n!*x^n = Sum_{n>=1} a(n+1)*x^n/n [Bowen, 1976].
a(n) = (-1)^(n-1) * det {| 1! 2! ... n! | 1 1! ... (n-1)! | 0 1 1! ... (n-2)! | ... | 0 ... 0 1 1! |}.
Gives the row sums of the triangle [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938; this triangle A089949. - Philippe Deléham, Dec 30 2003
L.g.f.: Sum_{n>=1} a(n)*x^n/n = log( Sum_{n>=0} n!*x^n ). - Paul D. Hanna, Sep 19 2007
G.f.: 1+x/(1-x/(1-2*x/(1-2*x/(1-3*x/(1-3*x/(1-4*x/(1-4*x/(1-...)))))))) (continued fraction). - Paul Barry, Oct 07 2008
a(n) = upper left term in M^(n-1), M = triangle A128175 as an infinite square production matrix (deleting the first "1"); as follows:
1, 1, 0, 0, 0, 0, ...
2, 2, 1, 0, 0, 0, ...
4, 4, 3, 1, 0, 0, ...
8, 8, 7, 4, 1, 0, ...
16, 16, 15, 11, 5, 1, ...
... (End)
O.g.f. satisfies: A(x) = x - x*A(x) + A(x)^2 + x^2*A'(x). - Paul D. Hanna, Jul 30 2011
Let A(x) be the g.f.; then
A(x) = 1/Q(0), where Q(k) = x + 1 + x*k - (k+2)*x/Q(k+1).
A(x) = (1-1/U(0))/x, when U(k) = 1 + x*(2*k+1)/(1 - 2*x*(k+1)/(2*x*(k+1) + 1/U(k+1))). (End)
Continued fractions:
G.f.: 1 - G(0)/2, where G(k) = 1 + 1/(1 - x*(2*k+2)/(x*(2*k+2) - 1 + x*(2*k+2)/G(k+1))).
G.f.: (x/2)*G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1/2) + 1/G(k+1))).
G.f.: x*G(0), where G(k) = 1 - x*(k+1)/(x - 1/G(k+1)).
G.f.: 1 - 1/G(0), where G(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+1)/(x*(k+1) - 1/G(k+1)))).
G.f.: x*W(0), where W(k) = 1 - x*(k+1)/(x*(k+1) - 1/(1 - x*(k+2)/(x*(k+2) - 1/W(k+1)))).
(End)
a(n) = A233824(n-1) if n > 0. (Proof. Set b(n) = A233824(n), so that b(n) = n*n! - Sum_{k=1..n-1} k!*b(n-k). To get a(n+1) = b(n) for n >= 0, induct on n, use (n+1)! = n*n! + n!, and replace k with k+1 in the sum.) - Jonathan Sondow, Dec 19 2013
a(n) ~ n! * (1 - 2/n - 1/n^2 - 5/n^3 - 32/n^4 - 253/n^5 - 2381/n^6 - 25912/n^7 - 319339/n^8 - 4388949/n^9 - 66495386/n^10), for coefficients see A260503. - Vaclav Kotesovec, Jul 27 2015
G.f.: 1 + x/(1 + x - 2*x/(1 + 2*x - 3*x/(1 + 3*x - 4*x/(1 + 4*x - ...)))). Cf. A000698.
G.f.: 1/(1 - x/(1 + x - x/(1 - 2*x/(1 - 2*x/(1 - 3*x/(1 - 3*x/(1 - 4*x/(1 - 4*x/(1 - ...))))))))). (End)
a(n) = R_1(n-2, 0) for n > 1 with a(0) = a(1) = 1 where R_1(n, q) = (q+2)*R_1(n-1, q+1) + Sum_{j=0..q} R_1(n-1, j) for n > 0, q >= 0 with R_1(0, q) = 1 for q >= 0.
a(n) = R_2(n-2, 0) for n > 1 with a(0) = a(1) = 1 where R_2(n, q) = Sum_{j=0..q+1} binomial(q+2, j+1)*R_2(n-1, j) for n > 0, q >= 0 with R_2(0, q) = 1 for q >= 0.
Here we can rewrite a(n) = R_1(n-2, 0) as a(n) = Sum_{j=0..2^(n-2) - 1} b_1(j) where b_1(2^m*(2k+1)) = (m+1)*b_1(2^(m+1)*k) + Sum_{j=0..m} b_1(2^j*k) for m >= 0, k >= 0 with b_1(0) = 1. We can also rewrite a(n) = R_2(n-2, 0) as a(n) = Sum_{j=0..2^(n-2) - 1} b_2(j) where b_2(2^m*(2k+1)) = Sum_{j=0..m} binomial(m+2, j+1)*b_2(2^j*k) for m >= 0, k >= 0 with b_2(0) = 1. FindStat provides a sequence of mappings between b_1(n) (or b_2(n)) and this sequence starting from collection [Permutations] where we take only irreducible permutations (see Links section for illustration). (End)
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EXAMPLE
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G.f. = 1 + x + x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 461*x^6 + 3447*x^7 + 29093*x^8 + ...
A permutation p in [n] (where n >= 0) is reducible if there exists an i in 1..n-1 such that for all j in the range 1..i and all k in the range i+1..n it is true that p(j) < p(k). (Note that a range a..b includes a and b.) If such an i exists we say that i splits the permutation at i.
Examples:
* () is not reducible since there is no index i which splits (). (=> a(0) = 1)
* (1) is not reducible since there is no index i which splits (1). (=> a(1) = 1)
* (1, 2) is reducible since index 1 splits (1, 2) as p(1) < p(2).
* (2, 1) is not reducible since at the only potential splitting point i = 1 we have p(1) > p(2). (=> a(2) = 1)
* For n = 3 we have (1, 2, 3), (1, 3, 2), and (2, 1, 3) are reducible and (2, 3, 1), (3, 1, 2), and (3, 2, 1) are irreducible. (End)
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MAPLE
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INVERTi([seq(n!, n=1..20)]);
A003319 := proc(n) option remember; n! - add((n-j)!*A003319(j), j=1..n-1) end;
series(2 - 1/hypergeom([1, 1], [], x), x=0, 50); # Mark van Hoeij, Apr 18 2013
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MATHEMATICA
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a[n_] := a[n] = n! - Sum[k!*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 11 2011, after given formula *)
CoefficientList[Assuming[Element[x, Reals], Series[2-E^(1/x)* x/ExpIntegralEi[1/x], {x, 0, 20}]], x] (* Vaclav Kotesovec, Mar 07 2014 *)
a[ n_] := If[ n < 2, 1, a[n] = (n - 2) a[n - 1] + Sum[ a[k] a[n - k], {k, n - 1}]]; (* Michael Somos, Feb 23 2015 *)
Table[SeriesCoefficient[1 + x/(1 + ContinuedFractionK[-Floor[(k + 2)/2]*x, 1, {k, 1, n}]), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 29 2017 *)
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PROG
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(PARI) {a(n) = my(A); if( n<1, 1, A = vector(n); A[1] = 1; for( k=2, n, A[k] = (k - 2) * A[k-1] + sum( j=1, k-1, A[j] * A[k-j])); A[n])}; /* Michael Somos, Jul 24 2011 */
(PARI) {if(n<1, 1, a(n)=local(A=x); for(i=1, n, A=x-x*A+A^2+x^2*A' +x*O(x^n)); polcoeff(A, n))} /* Paul D. Hanna, Jul 30 2011 */
(Sage)
R, C = [1], [1] + [0] * (len - 1)
for n in range(1, len):
for k in range(n, 0, -1):
C[k] = C[k - 1] * k
C[0] = -sum(C[k] for k in range(1, n + 1))
R.append(-C[0])
return R
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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Additional comments from Marcelo Aguiar (maguiar(AT)math.tamu.edu), Mar 28 2002
Added a(0)=0 (some of the formulas may now need adjusting). - N. J. A. Sloane, Sep 12 2012
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STATUS
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approved
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