A274888 Triangle read by rows: the q-analog of the swinging factorial which is defined as q-multinomial([floor(n/2), n mod 2, floor(n/2)]).

1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 2, 4, 5, 6, 5, 4, 2, 1, 1, 1, 2, 3, 3, 3, 3, 2, 1, 1, 1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1, 1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 2, 4, 7, 12, 17, 24, 31, 39, 45, 51, 54, 56, 54, 51, 45, 39, 31, 24, 17, 12, 7, 4, 2, 1

Offset

0,6

Comments

The q-swing_factorial(n) is a univariate polynomial over the integers with degree floor((n+1)/2)^2 + ((n+1) mod 2) and at least floor(n/2) irreducible factors.

Evaluated at q=1 q-swing_factorial(n) gives the swinging factorial A056040(n).

Combinatorial interpretation: The definition of an orbital system is given in A232500 and in the link 'Orbitals'. The number of orbitals over n sectors is counted by the swinging factorial.

The major index of an orbital is the sum of the positions of steps which are immediately followed by a step with strictly smaller value. This statistic is an extension of the major index statistic given in A063746 which reappears in the even numbered rows here. This reflects the fact that the swinging factorial can be seen as an extension of the central binomial. As in the case of the central binomial also in the case of the swinging factorial the major index coincides with its q-analog.

Links

Table of n, a(n) for n=0..89.

Peter Luschny, Orbitals

Formula

q_swing_factorial(n) = q_factorial(n)/q_factorial(floor(n/2))^2.

q_swing_factorial(n) = q_binomial(n-eta(n),floor((n-eta(n))/2))*q_int(n)^eta(n) with eta(n) = (1-(-1)^n)/2.

Recurrence: q_swing_factorial(0,q) = 1 and for n>0 q_swing_factorial(n,q) = r*q_swing_factorial(n-1,q) with r = (1+q^(n/2))/[n/2;q] if n is even else r = [n;q]. Here [a;q] are the q_brackets.

The generating polynomial for row n is P_n(p) = ((p^(floor(n/2)+1)-1)/(p-1))^((1-(-1)^n)/2)*Product_{i=0..floor(n/2)-1}((p^(n-i)-1)/(p^(i+1)-1)).

Example

The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^2 + q + 1)
[5] (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q + 1) * (q^2 - q + 1) * (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
The coefficients of the polynomials start:
[n] [k=0,1,2,...] [row sum]
[0] [1] [1]
[1] [1] [1]
[2] [1, 1] [2]
[3] [1, 2, 2, 1] [6]
[4] [1, 1, 2, 1, 1] [6]
[5] [1, 2, 4, 5, 6, 5, 4, 2, 1] [30]
[6] [1, 1, 2, 3, 3, 3, 3, 2, 1, 1] [20]
[7] [1, 2, 4, 7, 10, 13, 16, 17, 17, 16, 13, 10, 7, 4, 2, 1] [140]
[8] [1, 1, 2, 3, 5, 5, 7, 7, 8, 7, 7, 5, 5, 3, 2, 1, 1] [70]
T(5, 4) = 6 because the 2 orbitals [-1,-1,1,1,0] and [-1,0,1,1,-1] have at position 4 and the 4 orbitals [0,-1,1,-1,1], [1,-1,0,-1,1], [1,-1,1,-1,0] and [1,0,1,-1,-1] at positions 1 and 3 a down step.

Maple

QSwingFactorial_coeffs := proc(n) local P, a, b;
a := mul((p^(n-i)-1)/(p^(i+1)-1), i=0..iquo(n, 2)-1);
b := ((p^(iquo(n, 2)+1)-1)/(p-1))^((1-(-1)^n)/2);
P := simplify(a*b); seq(coeff(P, p, j), j=0..degree(P)) end:
for n from 0 to 9 do print(QSwingFactorial_coeffs(n)) od;
# Alternatively (recursive):
with(QDifferenceEquations):
QSwingRec := proc(n, q) local r; if n = 0 then return 1 fi:
if irem(n, 2) = 0 then r := (1+q^(n/2))/QBrackets(n/2, q)
else r := QBrackets(n, q) fi; r*QSwingRec(n-1, q) end:
Trow := proc(n) expand(QSimplify(QSwingRec(n, q)));
seq(coeff(%, q, j), j=0..degree(%)) end: seq(Trow(n), n=0..10);

Mathematica

p[n_] := QFactorial[n, q] / QFactorial[Quotient[n, 2], q]^2
Table[CoefficientList[p[n] // FunctionExpand, q], {n, 0, 9}] // Flatten

Prog

(Sage)
from sage.combinat.q_analogues import q_factorial
def q_swing_factorial(n, q=None):
    return q_factorial(n)//q_factorial(n//2)^2
for n in (0..8): print(q_swing_factorial(n).list())
(Sage) # uses[unit_orbitals from A274709]
# Brute force counting
def orbital_major_index(n):
    S = [0]*(((n+1)//2)^2 + ((n+1) % 2))
    for u in unit_orbitals(n):
        L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)]
        # i+1 because u is 0-based whereas convention assumes 1-base.
        S[sum(L)] += 1
    return S
for n in (0..9): print(orbital_major_index(n))

Crossrefs

Cf. A056040 (row sums), A274887 (q-factorial), A063746 (q-central_binomial).

Other orbital statistics: A241477 (first zero crossing), A274706 (absolute integral), A274708 (peaks), A274709 (max. height), A274710 (number of turns), A274878 (span), A274879 (returns), A274880 (restarts), A274881 (ascent).

Sequence in context: A156054 A030616 A066422 * A239702 A202084 A337763

Adjacent sequences: A274885 A274886 A274887 * A274889 A274890 A274891

Keyword

nonn,tabf

Author

Peter Luschny, Jul 19 2016

Status

approved