A261592 9-Modular Catalan Numbers C_{n,9}.

1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 58773, 207907, 742220, 2670564, 9674496, 35256723, 129164090, 475418625, 1757248194, 6519768464, 24272733060, 90648139140, 339497371575, 1274821281747, 4798525000860, 18102238168134, 68430875696534

Offset: 0

Author

Nickolas Hein, Aug 25 2015

Keywords

• nonn

Comments

Definition: Given a primitive k-th root of unity w, a binary operation a*b=a+wb, and sufficiently general fixed complex numbers x_0, ..., x_n, the k-modular Catalan numbers C_{n,k} enumerate parenthesizations of x_0*x_1*...*x_n that give distinct values.

Theorem: C_{n,k} enumerates the following objects:

(1) binary trees with n internal nodes avoiding a certain subtree (i.e., comb_k^{+1}),

(2) plane trees with n+1 nodes whose non-root nodes have degree less than k,

(3) Dyck paths of length 2n avoiding a down-step followed immediately by k consecutive up-steps,

(4) partitions with n nonnegative parts bounded by the staircase partition (n-1,n-2,...,1,0) such that each positive number appears fewer than k times,

(5) standard 2-by-n Young tableaux whose top row avoids contiguous labels of the form i,j+1,j+2,...,j+k for all i

(6) permutations of {1,2,...,n} avoiding 1-3-2 and 23...(k+1)1.

Examples

			The Catalan number C_10=16796 counts the parenthesizations of x_1*...*x_11 where * is arbitrary. Defining * and w as above and writing x_i compactly as xi, we have x1*(x2*(x3*(x4*(x5*(x6*(x7*(x8*(x9*(x10*(x11)))))))))) = x1+wx2+w^2x3+w^3x4+w^4x5+w^5x6+w^6x7+w^7x8+w^8x9+x10+wx11 = x1*(x2*(x3*(x4*(x5*(x6*(x7*(x8*(x9*(x10)))))))))*(x11). For n=10 and k=9, these are the only parenthesizations that give the same value for x1*...*x11, so C_{10,9}=16796-1=16795.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Nickolas Hein, Jia Huang, Modular Catalan Numbers, arXiv:1508.01688 [math.CO], 2015-2016.

Crossrefs

Column k=9 of A295679.

C_{n,1} is the all 1's sequence A000012. For C_{n,k} with k=2,3,4 see A011782, A005773, A159772. For k=5,6,7,8 see A261588, A261589, A261590, A261591.

Cf. A291823.

Programs

Mathematica

terms = 30; col[k_] := Module[{G}, G = InverseSeries[x*(1 - x)/(1 - x^k) + O[x]^terms, x]; CoefficientList[1/(1 - G), x]];
col[9] (* Jean-François Alcover, Dec 05 2017, after Andrew Howroyd *)

PARI

Vec(1/(1-serreverse(x*(1-x)/(1-x^9) + O(x*x^25)))) \\ Andrew Howroyd, Nov 29 2017

Sage

def C(k):
    print(1)
    for n in range(1,51):
        f = ((1-x^k)/(1-x))^n # ((x+1)^2-x^2*(x/(x+1))^(k-2))^n
        f = f.simplify_full()
        C = 0
        for i in range(n):
            C = C + (n-i)*f.coefficient(x,i)/n
        print(C)
time C(9)

Formula

sum( 1<=l<=n, (l/n)sum( m_1+...+m_k=n and m_2+2m_3+...+(k-1)m_k=n-l , MC(n;m_1,...,m_k) ) ), where MC(n;m_1,...,m_k) is the multinomial coefficient associated to the multiset (m_1,...,m_k).

G.f.: 1/(1-x*G(x)) where G(x) is g.f. of A291823. - Andrew Howroyd, Nov 29 2017

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