A259472 -id:A259472 - cp's OEIS frontend

A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.

1, 1, 1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, 5201061455, 73943424413, 1123596277863, 18176728317413, 311951144828863, 5661698774848621, 108355864447215063, 2181096921557783605, 46066653228356851631, 1018705098450570562877

Offset: 0

N. J. A. Sloane

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.
Also the left border of triangle A144107, with row sums = n!. - Gary W. Adamson, Sep 11 2008
Hankel transform is A059332. Hankel transform of aerated sequence is A137704(n+1). - Paul Barry, Oct 07 2008
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

			G.f. = 1 + x + x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 461*x^6 + 3447*x^7 + 29093*x^8 + ...
From _Peter Luschny_, Aug 03 2022: (Start)
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)

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).

Seiichi Manyama, Table of n, a(n) for n = 0..449 (first 102 terms from T. D. Noe)
Marcelo Aguiar and Frank Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, arXiv:math/0203282 [math.CO], 2002-2005.
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. .
M. Aguiar and A. Lauve, Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.
M. Aguiar and A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, 2014.
Marcelo Aguiar and Swapneel Mahajan, On the Hadamard product of Hopf monoids, 2013.
Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, and Mike Zabrocki, When are Hopf algebras determined by integer sequences?, arXiv:2505.06941 [math.CO], 2025. See p. 10.
Joerg Arndt, Generating Random Permutations, PhD thesis, Australian National University, Canberra, Australia, (2010).
Joerg Arndt, Matters Computational (The Fxtbook), p. 281
Roland Bacher and Christophe Reutenauer, Number of right ideals and a q-analogue of indecomposable permutations, arXiv preprint arXiv:1511.00426 [math.CO], 2015.
Paul Barry, A note on number triangles that are almost their own production matrix, arXiv:1804.06801 [math.CO], 2018.
Julien Berestycki, Eric Brunet and Zhan Shi, How many evolutionary histories only increase fitness?, arXiv preprint arXiv:1304.0246 [math.PR], 2013.
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
E. W. Bowen and N. J. A. Sloane, Correspondence, 1976.
Glenn Bruda, Asymptotic expansions for the reciprocal Hardy-Littlewood logarithmic integrals, arXiv:2412.19866 [math.CO], 2024. See pages 1, 7.
David Callan, Counting Stabilized-Interval-Free Permutations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.8.
Peter Cameron's Blog, The symmetric group, 11, Posted 09/04/2011.
Mahir Bilen Can, Luke Nelson, and Kevin Treat, A Catalanization Map on the Symmetric Group, Enum. Comb. Appl. (2022) Vol. 2, No. 4, #S4PP7.
Daniel Chen and Sebastian Ohlig, Associated Permutations of Complete Non-Ambiguous Trees and Zubieta's Conjecture, arXiv:2210.11117 [math.CO], 2022.
Fan Chung and R. L. Graham, Primitive juggling sequences, Am. Math. Monthly 115 (3) (2008) 185-194.
L. Comtet, Sur les coefficients de l'inverse de la série formelle Sum n! t^n, Comptes Rend. Acad. Sci. Paris, A 275 (1972), 569-572.
L. Comtet, Series inversions, C. R. Acad. Sc. Paris, t. 275 (25 septembre 1972), 569-572. (Annotated scanned copy)
Xinle Dai, Jordan Long, and Karen Yeats, Subdivergence-free gluings of trees, arXiv:2106.07494 [math.CO], 2021.
M. A. Deryagina and A. D. Mednykh, On the enumeration of circular maps with given number of edges, Siberian Mathematical Journal, 54, No. 6, 2013, 624-639.
Stoyan Dimitrov, Sorting by shuffling methods and a queue, arXiv:2103.04332 [math.CO], 2021.
J. D. Dixon, The probability of generating the symmetric group, Math. Z. 110 (1969) 199-205.
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
G. Duchamp, F. Hivert, and J.-Y. Thibon, Noncommutative symmetric functions VI: Free quasi-symmetric functions and related algebras, arXiv:math/0105065 [math.CO], 2001.
G. A. Edgar, Transseries for beginners, arXiv:0801.4877 [math.RA], 2008-2009.
Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, Catalan-Spitzer permutations, arXiv:2310.06288 [math.CO], 2023. See p. 19.
Jesse Elliott, Asymptotic expansions of the prime counting function, arXiv:1809.06633 [math.NT], 2018.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 90.
A. L. L. Gao, S. Kitaev, and P. B. Zhang. On pattern avoiding indecomposable permutations, arXiv:1605.05490 [math.CO], 2016.
I. M. Gessel and R. P. Stanley Algebraic Enumeration (See pages 7-8 for generating function.)
R. K. Guy, Letter to N. J. A. Sloane, Mar 1974
P. Hegarty and A. Martinsson, On the existence of accessible paths in various models of fitness landscapes, arXiv preprint arXiv:1210.4798 [math.PR], 2012. - From _N. J. A. Sloane_, Jan 01 2013
V. Jelínek and P. Valtr, Splittings and Ramsey Properties of Permutation Classes, arXiv preprint arXiv:1307.0027 [math.CO], 2013.
B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.
A. King, Generating indecomposable permutations, Discrete Math., 306 (2006), 508-518.
Y. Koh and S. Ree, Connected permutation graphs, Discrete Math. 307 (2007), 2628-2635.
M. K. Krotter, I. C. Christov, J. M. Ottino and R. M. Lueptow, Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations, arXiv preprint arXiv:1208.2052 [physics.flu-dyn], 2012. - From _N. J. A. Sloane_, Dec 25 2012
Ajay Kumar and Chanchal Kumar, Monomial ideals induced by permutations avoiding patterns, 2018.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.
Steven Linton, James Propp, Tom Roby, and Julian West, Equivalence Classes of Permutations under Various Relations Generated by Constrained Transpositions, Journal of Integer Sequences, Vol. 15 (2012), #12.9.1.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, arXiv:1103.4936 [math.CO], 2011.
R. J. Martin and M. J. Kearney, An exactly solvable self-convolutive recurrence, Aequat. Math., 80 (2010), 291-318. see p. 292.
Richard J. Martin and Michael J. Kearney, Integral representation of certain combinatorial recurrences, Combinatorica: 35:3 (2015), 309-315.
Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions (2008); arXiv:0806.3682 [math.CO]. Discrete Math. 310 (2010), no. 24, 3584-3606.
P. Ossona de Mendez and P. Rosenstiehl, Transitivity and connectivity of permutations, Combinatorica, 24 (No. 3, 2004), 487-501.
L. Panaitopol, A formula for pi(x) applied to a result of Koninck-Ivić, Nieuw Arch. Wisk. 5/1 55-56 (2000).
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.
S. Poirier and C. Reutenauer, Algèbres Hopf de tableaux, Ann. Sci. Math. Quebec 19 (95), no. 1, 79-90.
N. Reading Noncrossing partitions and the shard intersection order. J. Algebraic Combin. 33 (2011), no. 4. arXiv preprint arXiv:0909.3288, [math.CO], 2009.
Herman P. Robinson, Letter to N. J. A. Sloane, Nov 19 1973.
R. P. Stanley, The Descent Set and Connectivity Set of a Permutation, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.8.
Timothy R. Walsh, Space-efficient generation of nonisomorphic maps and hypermaps, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.3.
Marcel Wienöbst, Max Bannach, and Maciej Liśkiewicz, Polynomial-Time Algorithms for Counting and Sampling Markov Equivalent DAGs with Applications, arXiv:2205.02654 [cs.LG], 2022.
Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 7.

See A167894 for another version.
Bisections give A272656, A272657.
Row sums of A111184 and A089949.
Leading diagonal of A059438. A diagonal of A263484.
Cf. A051296, A084938, A074664, A113869, A144107.
Cf. A260503, A259472, A261214, A261239, A261253, A261254.
Cf. A090238, A000698, A356291 (reducible permutations).
Column k=0 of A370380 and A370381 (without pair of initial terms and with different offset).

INVERTi([seq(n!,n=1..20)]);
A003319 := proc(n) option remember; n! - add((n-j)!*A003319(j), j=1..n-1) end;
[seq(A003319(n), n=0..50)]; # N. J. A. Sloane, Dec 28 2011
series(2 - 1/hypergeom([1,1], [], x), x=0,50); # Mark van Hoeij, Apr 18 2013

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 *)

{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 */

{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 */

def A003319_list(len):
    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
print(A003319_list(21))  # Peter Luschny, Feb 19 2016

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! |}.
INVERTi transform of factorial numbers, A000142 starting from n=1. - Antti Karttunen, May 30 2003
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
a(n+1) = Sum_{k=0..n} A089949(n,k). - Philippe Deléham, Oct 16 2006
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) = -Sum_{i=0..n} (-1)^i*A090238(n, i) for n > 0. - Peter Luschny, Mar 13 2009
From Gary W. Adamson, Jul 14 2011: (Start)
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
From Sergei N. Gladkovskii, Jun 24 2012: (Start)
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)
From Sergei N. Gladkovskii, Aug 03 2013: (Start)
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
For n>0, a(n) = (A059439(n) - A259472(n))/2. - Vaclav Kotesovec, Aug 03 2015
From Peter Bala, May 23 2017: (Start)
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)
Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - Mikhail Kurkov, Apr 26 2024

More terms from Michael Somos, Jan 26 2000
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
Edited and set a(0) = 1 by Peter Luschny, Aug 03 2022

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A261214 Coefficients in an asymptotic expansion of sequence A259472.

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A261239 Coefficients in an asymptotic expansion of A259472 in falling factorials.

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A003319 Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.

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A260503 Coefficients in an asymptotic expansion of sequence A003319.

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A261253 Coefficients in an asymptotic expansion of sequence A261239.

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A261254 Coefficients in an asymptotic expansion of A261239 in falling factorials.

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A059439 A diagonal of A059438.

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