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Identifier

St001786: Dyck paths ⟶ ℤ

Values

[1,0] => 1

[1,0,1,0] => 1

[1,1,0,0] => 1

[1,0,1,0,1,0] => 1

[1,0,1,1,0,0] => 1

[1,1,0,0,1,0] => 1

[1,1,0,1,0,0] => 2

[1,1,1,0,0,0] => 1

[1,0,1,0,1,0,1,0] => 1

[1,0,1,0,1,1,0,0] => 1

[1,0,1,1,0,0,1,0] => 1

[1,0,1,1,0,1,0,0] => 2

[1,0,1,1,1,0,0,0] => 1

[1,1,0,0,1,0,1,0] => 1

[1,1,0,0,1,1,0,0] => 1

[1,1,0,1,0,0,1,0] => 2

[1,1,0,1,0,1,0,0] => 4

[1,1,0,1,1,0,0,0] => 3

[1,1,1,0,0,0,1,0] => 1

[1,1,1,0,0,1,0,0] => 2

[1,1,1,0,1,0,0,0] => 3

[1,1,1,1,0,0,0,0] => 1

[1,0,1,0,1,0,1,0,1,0] => 1

[1,0,1,0,1,0,1,1,0,0] => 1

[1,0,1,0,1,1,0,0,1,0] => 1

[1,0,1,0,1,1,0,1,0,0] => 2

[1,0,1,0,1,1,1,0,0,0] => 1

[1,0,1,1,0,0,1,0,1,0] => 1

[1,0,1,1,0,0,1,1,0,0] => 1

[1,0,1,1,0,1,0,0,1,0] => 2

[1,0,1,1,0,1,0,1,0,0] => 4

[1,0,1,1,0,1,1,0,0,0] => 3

[1,0,1,1,1,0,0,0,1,0] => 1

[1,0,1,1,1,0,0,1,0,0] => 2

[1,0,1,1,1,0,1,0,0,0] => 3

[1,0,1,1,1,1,0,0,0,0] => 1

[1,1,0,0,1,0,1,0,1,0] => 1

[1,1,0,0,1,0,1,1,0,0] => 1

[1,1,0,0,1,1,0,0,1,0] => 1

[1,1,0,0,1,1,0,1,0,0] => 2

[1,1,0,0,1,1,1,0,0,0] => 1

[1,1,0,1,0,0,1,0,1,0] => 2

[1,1,0,1,0,0,1,1,0,0] => 2

[1,1,0,1,0,1,0,0,1,0] => 4

[1,1,0,1,0,1,0,1,0,0] => 8

[1,1,0,1,0,1,1,0,0,0] => 6

[1,1,0,1,1,0,0,0,1,0] => 3

[1,1,0,1,1,0,0,1,0,0] => 6

[1,1,0,1,1,0,1,0,0,0] => 9

[1,1,0,1,1,1,0,0,0,0] => 4

[1,1,1,0,0,0,1,0,1,0] => 1

[1,1,1,0,0,0,1,1,0,0] => 1

[1,1,1,0,0,1,0,0,1,0] => 2

[1,1,1,0,0,1,0,1,0,0] => 4

[1,1,1,0,0,1,1,0,0,0] => 3

[1,1,1,0,1,0,0,0,1,0] => 3

[1,1,1,0,1,0,0,1,0,0] => 6

[1,1,1,0,1,0,1,0,0,0] => 9

[1,1,1,0,1,1,0,0,0,0] => 6

[1,1,1,1,0,0,0,0,1,0] => 1

[1,1,1,1,0,0,0,1,0,0] => 2

[1,1,1,1,0,0,1,0,0,0] => 3

[1,1,1,1,0,1,0,0,0,0] => 4

[1,1,1,1,1,0,0,0,0,0] => 1

[1,0,1,0,1,0,1,0,1,0,1,0] => 1

[1,0,1,0,1,0,1,0,1,1,0,0] => 1

[1,0,1,0,1,0,1,1,0,0,1,0] => 1

[1,0,1,0,1,0,1,1,0,1,0,0] => 2

[1,0,1,0,1,0,1,1,1,0,0,0] => 1

[1,0,1,0,1,1,0,0,1,0,1,0] => 1

[1,0,1,0,1,1,0,0,1,1,0,0] => 1

[1,0,1,0,1,1,0,1,0,0,1,0] => 2

[1,0,1,0,1,1,0,1,0,1,0,0] => 4

[1,0,1,0,1,1,0,1,1,0,0,0] => 3

[1,0,1,0,1,1,1,0,0,0,1,0] => 1

[1,0,1,0,1,1,1,0,0,1,0,0] => 2

[1,0,1,0,1,1,1,0,1,0,0,0] => 3

[1,0,1,0,1,1,1,1,0,0,0,0] => 1

[1,0,1,1,0,0,1,0,1,0,1,0] => 1

[1,0,1,1,0,0,1,0,1,1,0,0] => 1

[1,0,1,1,0,0,1,1,0,0,1,0] => 1

[1,0,1,1,0,0,1,1,0,1,0,0] => 2

[1,0,1,1,0,0,1,1,1,0,0,0] => 1

[1,0,1,1,0,1,0,0,1,0,1,0] => 2

[1,0,1,1,0,1,0,0,1,1,0,0] => 2

[1,0,1,1,0,1,0,1,0,0,1,0] => 4

[1,0,1,1,0,1,0,1,0,1,0,0] => 8

[1,0,1,1,0,1,0,1,1,0,0,0] => 6

[1,0,1,1,0,1,1,0,0,0,1,0] => 3

[1,0,1,1,0,1,1,0,0,1,0,0] => 6

[1,0,1,1,0,1,1,0,1,0,0,0] => 9

[1,0,1,1,0,1,1,1,0,0,0,0] => 4

[1,0,1,1,1,0,0,0,1,0,1,0] => 1

[1,0,1,1,1,0,0,0,1,1,0,0] => 1

[1,0,1,1,1,0,0,1,0,0,1,0] => 2

[1,0,1,1,1,0,0,1,0,1,0,0] => 4

[1,0,1,1,1,0,0,1,1,0,0,0] => 3

[1,0,1,1,1,0,1,0,0,0,1,0] => 3

[1,0,1,1,1,0,1,0,0,1,0,0] => 6

[1,0,1,1,1,0,1,0,1,0,0,0] => 9

[1,0,1,1,1,0,1,1,0,0,0,0] => 6

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$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + q^{2}$

$F_{4} = 8\ q + 3\ q^{2} + 2\ q^{3} + q^{4}$

$F_{5} = 16\ q + 8\ q^{2} + 6\ q^{3} + 5\ q^{4} + 4\ q^{6} + q^{8} + 2\ q^{9}$

$F_{6} = 32\ q + 20\ q^{2} + 16\ q^{3} + 15\ q^{4} + 2\ q^{5} + 12\ q^{6} + 6\ q^{8} + 8\ q^{9} + 2\ q^{10} + 7\ q^{12} + 3\ q^{16} + 6\ q^{18} + q^{24} + 2\ q^{27}$

$F_{7} = 64\ q + 48\ q^{2} + 40\ q^{3} + 41\ q^{4} + 6\ q^{5} + 38\ q^{6} + 19\ q^{8} + 22\ q^{9} + 9\ q^{10} + 25\ q^{12} + 4\ q^{15} + 13\ q^{16} + 24\ q^{18} + 5\ q^{20} + 14\ q^{24} + 2\ q^{25} + 10\ q^{27} + 4\ q^{30} + 4\ q^{32} + 11\ q^{36} + 4\ q^{40} + 3\ q^{48} + 2\ q^{50} + 8\ q^{54} + q^{60} + 2\ q^{64} + 3\ q^{72} + 2\ q^{81} + q^{96}$

Description

The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order.
Alternatively, remark that the monomials of the polynomial

$\prod_{k=1}^n (z_1+\dots +z_k)$ are in bijection with Dyck paths, regarded as superdiagonal paths, with

$n$ east steps: the exponent of

$z_i$ is the number of north steps before the

$i$ -th east step, see [2]. Thus, this statistic records the coefficients of the monomials.
A formula for the coefficient of

$z_1^{a_1}\dots z_n^{a_n}$ is provided in [3]:

$c_{(a_1,\dots,a_n)} = \prod_{k=1}^{n-1} \frac{n-k+1 - \sum_{i=k+1}^n a_i}{a_k!}.$
This polynomial arises in a partial symmetrization process as follows, see [1]. For

$w\in\frak{S}_n$ , let

$w\cdot F(x_1,\dots,x_n)=F(x_{w(1)},\dots,x_{w(n)})$ . Furthermore, let

$G(\mathbf{x},\mathbf{z}) = \prod_{k=1}^n\frac{x_1z_1+x_2z_2+\cdots+x_kz_k}{x_k-x_{k+1}}.$
Then

$\sum_{w\in\frak{S}_{n+1}}w\cdot G = \prod_{k=1}^n (z_1+\dots +z_k)$ .

References

[1] Amdeberhan, T. Counting monomials and the Catalan numbers MathOverflow:420887
[2] The coefficients in the expansion x_1(x_1 + x_2)...(x_1 + x_2 + ... + x_n), given row by row. OEIS:A347917
[3] Fried, S. On the coefficients of the distinct monomials in the expansion of $x_1(x_1+x_2)\cdots (x_1+x_2+\cdots +x_n)$ arXiv:2111.07331

Code

def statistic(D):
    n = D.semilength()
    a = []
    i = 0
    for e in D:
        if e == 0:
            a.append(i)
            i = 0
        else:
            i += 1
    return prod((n-k-sum(a[k+1:]))/factorial(a[k]) for k in range(len(a)-1))

# alternative, direct
@cached_function
def G_sym(n):
    Z = PolynomialRing(QQ, n-1, "z").gens()
    return prod(sum(Z[:k]) for k in range(1, n))

def to_dyck(m):
    D = []
    e = m.exponents()
    assert len(e) == 1, "should be a single monomial, but is %s" % m
    for N in e[0]:
        D.extend([1]*N)
        D.append(0)
    return DyckWord(D)

def statistics(n):
    p = G_sym(n)
    s = {}
    for c, m in p:
        s[to_dyck(m)] = c
    return s

def statistic(D):
    return statistics(D.semilength()+1)[D]

# alternative, completely direct
@cached_function
def G_sym(n):
    R = PolynomialRing(QQ, n-1, "z")
    S = PolynomialRing(R, n, "x")
    X = S.gens()
    p = [x*z for z, x in zip(R.gens(), X)]
    G = prod(sum(p[:k])/(X[k-1]-X[k]) for k in range(1,n))
    G_s = 0
    for pi in Permutations(n):
        s = G.subs({X[i]: X[pi[i]-1] for i in range(n)})
        G_s += s
    return R(G_s)

def to_dyck(m):
    D = []
    e = m.exponents()
    assert len(e) == 1, "should be a single monomial, but is %s" % m
    for N in e[0]:
        D.extend([1]*N)
        D.append(0)
    return DyckWord(D)

def statistics(n):
    p = G_sym(n)
    s = {}
    for c, m in p:
        s[to_dyck(m)] = c
    return s

def statistic(D):
    return statistics(D.semilength()+1)[D]

Created

Apr 25, 2022 at 10:03 by Martin Rubey

Updated

Apr 26, 2022 at 17:11 by Martin Rubey