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A212851 Number of n X 4 arrays with rows being permutations of 0..3 and no column j greater than column j-1 in all rows. 13

%I #20 Aug 26 2019 04:22:36

%S 1,211,8983,271375,7225951,182199871,4479288703,108787179775,

%T 2626338801151,63217691436031,1519452489242623,36493601345048575,

%U 876167372044132351,21031868446675976191,504811062363654815743

%N Number of n X 4 arrays with rows being permutations of 0..3 and no column j greater than column j-1 in all rows.

%C Column 4 of A212855.

%C From _Petros Hadjicostas_, Aug 25 2019:

%C All formulas below follow from the theory in the documentation of array A309951.

%C We have Sum_{s = 0..A000041(4)} (-1)^s * A309951(4,s) * a(n-s) = 0, i.e., a(n) - 47*a(n-1) + 718*a(n-2) - 4416*a(n-3) + 10656*a(n-4) - 6912*a(n-5) = 0 for n >= 6. This is a consequence of Eq. (6) on p. 248 of Abramson and Promislow (1978).

%C Note that in _R. J. Mathar_'s formula a(n) = 24^n + 6^n - 3*12^n + 2*4^n - 1^n, the numbers 1, 4, 12, 6, and 24 (that are raised to the n-th power) are the multinomial coefficients of the A000041(4) = 5 integer partitions of 4: 4!/4! = 1, 4!/(1!3!) = 4, 12 = 4!/(1!1!2!), 6 = 4!/(2!2!), 24 = 4!/(1!1!1!1!).

%C Note also that these numbers appear also in the denominator of the _Colin Barker_'s g.f.: (1 - x)*(1 - 4*x)*(1 - 6*x)*(1 - 12*x)*(1 - 24*x) = 1 - 47*x + 718*x^2 - 4416*x^3 + 10656*x^4 - 6912*x^5.

%C (End)

%H R. H. Hardin, <a href="/A212851/b212851.txt">Table of n, a(n) for n = 1..210</a>

%H Morton Abramson and David Promislow, <a href="https://doi.org/10.1016/0097-3165(78)90012-2">Enumeration of arrays by column rises</a>, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6) on p. 248 (set t:=0).

%F Empirical: a(n) = 47*a(n-1) - 718*a(n-2) + 4416*a(n-3) - 10656*a(n-4) + 6912*a(n-5).

%F Empirical: a(n) = 24^n + 6^n - 3*12^n + 2*4^n - 1. _R. J. Mathar_, Jun 25 2012

%F Empirical g.f.: x*(1 + 164*x - 216*x^2 - 3744*x^3) / ((1 - x)*(1 - 4*x)*(1 - 6*x)*(1 - 12*x)*(1 - 24*x)). - _Colin Barker_, Jul 21 2018

%e Some solutions for n=3:

%e ..1..3..0..2....3..1..2..0....1..2..0..3....1..2..0..3....1..2..0..3

%e ..2..1..0..3....3..1..0..2....0..1..3..2....3..0..2..1....2..1..3..0

%e ..2..3..1..0....1..2..0..3....3..2..0..1....1..2..0..3....1..3..2..0

%Y Cf. A000041, A212850, A212852, A212853, A212854, A212855, A212856, A309951.

%K nonn

%O 1,2

%A _R. H. Hardin_, May 28 2012

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Last modified March 28 14:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)