A124302 Number of set partitions with at most 3 blocks; number of Dyck paths of height at most 4; dimension of space of symmetric polynomials in 3 noncommuting variables.

1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165, 3812798742494, 11438396227481

Offset: 0

Author

Mike Zabrocki, Oct 25 2006

Keywords

• nonn
• easy

Comments

Row sums of triangle in A056241. - Philippe Deléham, Oct 30 2006

Row sums of triangle in A147746. - Philippe Deléham, Dec 04 2008

Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...]. - Philippe Deléham, Dec 04 2008

Number of nonisomorphic graded posets with 0 and 1 and uniform Hasse graph of rank n with no 3-element antichain. (Uniform used in the sense of Retakh, Serconek and Wilson. Graded used in Stanley's sense that every maximal chain has the same length n.) - David Nacin, Feb 26 2012

Number of Dyck paths of length 2n and height at most 4. - Ira M. Gessel, Aug 06 2012

Examples

			There are 15 set partitions of {1,2,3,4}, only {{1},{2},{3},{4}} has more than 3 blocks, so a(4) = 14.
G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 41*x^5 + 122*x^6 + 365*x^7 + ...
		

References

• R. Stanley, Enumerative combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997, pp. 96-100.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Per Alexandersson and Frether Getachew, An involution on derangements, arXiv:2105.08455 [math.CO], 2021.
• N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005; Canad. J. Math. 60 (2008), no. 2, 266-296.
• Giulio Cerbai, Anders Claesson, and Luca Ferrari, Stack sorting with restricted stacks, arXiv:1907.08142 [cs.DS], 2019.
• S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
• Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universitat Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
• M. Hyatt and J. Remmel, The classification of 231-avoiding permutations by descents and maximum drop, arXiv preprint arXiv:1208.1052 [math.CO], 2012. - From _N. J. A. Sloane_, Dec 24 2012
• V. Jelinek, T. Mansour, and M. Shattuck, On multiple pattern avoiding set partitions, Adv. Appl. Math. 50 (2) (2013) 292-326, - From _N. J. A. Sloane_, Jan 01 2013
• Sergey Kitaev, Jeffrey Remmel and Mark Tiefenbruck, Marked mesh patterns in 132-avoiding permutations I, arXiv:1201.6243 [math.CO], 2012-2014 (Corollary 3, case k=4, pages 10-11). - From _N. J. A. Sloane_, May 09 2012
• Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II, Electronic Journal of Combinatorial Number Theory, Volume 15 #A16. (arXiv:1302.2274)
• László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
• V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Algebras Associated to Directed Graphs and Order Homology, arXiv:1010.6295 [math.RA], 2010-2011.
• Santiago Rojas-Rojas, Camila Muñoz, Edgar Barriga, Pablo Solano, Aldo Delgado, and Carla Hermann-Avigliano, Analytic Evolution for Complex Coupled Tight-Binding Models: Applications to Quantum Light Manipulation, arXiv:2310.12366 [quant-ph], 2023. See p. 12.
• M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables, Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
• Index entries for linear recurrences with constant coefficients, signature (4,-3).

Crossrefs

Cf. A056272, A123183, A001519, A000110, A008277, A038754, A048328, A068911, A211216.

Essentially the same as A007051.

Programs

Magma

I:=[1, 1, 2]; [n le 3 select I[n] else  4*Self(n-1) - 3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 25 2012

Maple

a:= proc(n); if n<3 then [1,1,2][n+1]; else 4*a(n-1)-3*a(n-2); fi; end:
# Mike Zabrocki, Oct 25 2006
with(GraphTheory): G:=PathGraph(5): A:= AdjacencyMatrix(G): nmax:=27; for n from 0 to 2*nmax do B(n):=A^n; b(n):=B(n)[1,1]; od: for n from 0 to nmax do a(n):=b(2*n) od: seq(a(n),n=0..nmax);
# Johannes W. Meijer, May 29 2010

Mathematica

a=Exp[x]-1; Range[0, 20]! CoefficientList[Series[1+a+a^2/2+a^3/6, {x,0,20}],x]
Join[{1}, LinearRecurrence[{4, -3}, {1, 2}, 20]] (* David Nacin, Feb 26 2012 *)
CoefficientList[Series[1 / (1 - x / (1 - x / (1 - x / (1 - x)))), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 25 2012 *)
Table[Sum[StirlingS2[n,k],{k,0,3}],{n,0,30}] (* Robert A. Russell, Mar 29 2018 *)

PARI

{a(n) = if( n<1, n==0, (3^(n-1) + 1) / 2)}; /* Michael Somos, Apr 03 2014 */

Python

def a(n, adict={0:1, 1:1, 2:2}):
    if n in adict:
        return adict[n]
    adict[n]=4*a(n-1) - 3*a(n-2)
    return adict[n] # David Nacin, Mar 04 2012

Formula

O.g.f.: (q^2 - 3*q + 1)/(3*q^2 - 4*q + 1) = Sum_{k=0..3} (q^k/Product_{i=1..k} (1-i*q)).

a(n) = 4*a(n-1) - 3*a(n-2); a(0) = 1, a(1) = 1, a(2) = 2, a(n) = Sum_{k=1..3} A008277(n,k).

Inverse binomial transform of A007581. - Philippe Deléham, Oct 30 2006

a(n) = Sum_{k=0..n} A056241(n,k), n >= 1. - Philippe Deléham, Oct 30 2006

a(0) = 1, a(n) = (3^(n-1) + 1)/2 for n >= 1, see A007051. - Philippe Deléham, Oct 30 2006

E.g.f.: (2 + 3*exp(x) + exp(3x))/6.

G.f.: 1 / (1 - x / (1 - x / (1 - x / (1 - x)))). - Michael Somos, May 03 2012

G.f.: 1 + x + 3*x^2*U(0)/2 where U(k) = 1 + 2/(3*3^k + 3*3^k/(1 - 18*x*3^k/ (9*x*3^k - 1/U(k+1)))); (continued fraction, 4-step). - Sergei N. Gladkovskii, Nov 01 2012

G.f.: 1+x*G(0) where G(k) = 1 + 2*x/( 1-2*x - x*(1-2*x)/(x + (1-2*x)*2/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 10 2012

a(n) = Sum_{k=0..3} Stirling2(n,k). - Robert A. Russell, Mar 29 2018

G.f.: Sum_{j=0..k} A248925(k,j)*x^j / Product_{j=1..k} 1-j*x with k=3. - Robert A. Russell, Apr 25 2018