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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.

%I A212851 #30 Aug 16 2025 21:10:51
%S A212851 1,211,8983,271375,7225951,182199871,4479288703,108787179775,
%T A212851 2626338801151,63217691436031,1519452489242623,36493601345048575,
%U A212851 876167372044132351,21031868446675976191,504811062363654815743,12116020140998121291775,290791139166323355287551
%N 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.
%C A212851 Column 4 of A212855.
%C A212851 From _Petros Hadjicostas_, Aug 25 2019: (Start)
%C A212851 All formulas below follow from the theory in the documentation of array A309951.
%C A212851 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 A212851 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 A212851 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. (End)
%H A212851 R. H. Hardin, <a href="/A212851/b212851.txt">Table of n, a(n) for n = 1..210</a>
%H A212851 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 A212851 Empirical: a(n) = 47*a(n-1) - 718*a(n-2) + 4416*a(n-3) - 10656*a(n-4) + 6912*a(n-5).
%F A212851 Empirical: a(n) = 24^n + 6^n - 3*12^n + 2*4^n - 1. _R. J. Mathar_, Jun 25 2012
%F A212851 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 A212851 Some solutions for n=3:
%e A212851 ..1..3..0..2....3..1..2..0....1..2..0..3....1..2..0..3....1..2..0..3
%e A212851 ..2..1..0..3....3..1..0..2....0..1..3..2....3..0..2..1....2..1..3..0
%e A212851 ..2..3..1..0....1..2..0..3....3..2..0..1....1..2..0..3....1..3..2..0
%t A212851 T[n_, k_] := T[n, k] = If[k == 0, 1, -Sum[Binomial[k, j]^n*(-1)^j*T[n, k-j], {j, 1, k}]];
%t A212851 a[n_] := T[n, 4];
%t A212851 Table[a[n], {n, 1, 15}] (* _Jean-François Alcover_, Apr 01 2024, after _Alois P. Heinz_ in A212855 *)
%Y A212851 Cf. A000041, A212850, A212852, A212853, A212854, A212855, A212856, A309951.
%K A212851 nonn
%O A212851 1,2
%A A212851 _R. H. Hardin_, May 28 2012