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A212853 Number of n X 6 arrays with rows being permutations of 0..5 and no column j greater than column j-1 in all rows. 13
1, 90921, 179781181, 191740223841, 164481310134301, 128645361626874561, 96426023622482278621, 70816637331790329140481, 51492108377805402906874141, 37256471170472317193421713601, 26890352949868734582700237312861 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Column 6 of A212855.

From Petros Hadjicostas, Sep 08 2019: (Start)

Let P_6 be the set of all lists b = (b_1, b_2, b_3, b_4, b_5, b_6) of integers b_i >= 0, i = 1, ..., 6, such that 1*b_1 + 2*b_2 + 3*b_3 + 4*b_4 + 5*b_5 + 6*b_6 = 6; i.e., P_6 is the set all integer partitions of 6. Then |P_6| = A000041(6) = 11.

From Eq. (6), p. 248, in Abramson and Promislow (1978), with t=0, we get a(n) = A212855(n,6) = Sum_{b in P_6} (-1)^(6-Sum_{j=1..6} b_j) * (b_1 + b_2 + b_3 + b_4 + b_5 + b_6)!/(b_1! * b_2! * b_3! * b_4! * b_5! * b_6!) * (6! / ((1!)^b_1 * (2!)^b_2 * (3!)^b_3 * (4!)^b_4 * (5!)^b_5 * (6!)^b_6))^n.

The integer partitions of 6 are listed on p. 831 of Abramowitz and Stegun (1964). We see that the corresponding multinomial coefficients 6! / ((1!)^b_1 * (2!)^b_2 * (3!)^b_3 * (4!)^b_4 * (5!)^b_5 * (6!)^b_6) are all distinct; that is, A070289(6) = A000041(6) = 11 and A309951(6,s) = A325305(6,s) for s = 0..11. (Compare with the comments for A212854.)

Using the information about partitions of 6 in Eq. (6) (with t=0), p. 248, of Abramson and Promislow (1978), we may derive the explicit equation for a(n) shown below.

Using standard results from the theory of difference equations (since the solution is known explicitly), we may derive R. H. Hardin's empirical recurrence. The recurrence is equivalent to Sum_{s = 0..11} (-1)^s * A325305(6,s) * a(n-s) = 0 for n >= 12.

(End)

LINKS

R. H. Hardin, Table of n, a(n) for n = 1..210

Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards (Applied Mathematics Series, 55), 1964; see pp. 831-832 for the multinomial coefficients of integer partitions of n = 1..10.

Morton Abramson and David Promislow, Enumeration of arrays by column rises, J. Combinatorial Theory Ser. A 24(2) (1978), 247-250; see Eq. (6) (with t=0), p. 248, and the comments above.

Wikipedia, Partition (number theory).

Wikipedia, Multinomial theorem.

FORMULA

Empirical: a(n) = 1602*a(n-1) - 929171*a(n-2) + 260888070*a(n-3) - 39883405500*a(n-4) + 3492052425000*a(n-5) - 177328940580000*a(n-6) + 5153150631600000*a(n-7) - 82577533320000000*a(n-8) + 669410956800000000*a(n-9) - 2224399449600000000*a(n-10) + 1632586752000000000*a(n-11) for n >= 12. [It is correct; see the comments above.]

a(n) = -1 + 2*6^n + 2*15^n + 20^n - 3*30^n - 6*60^n - 90^n + 4*120^n + 6*180^n - 5*360^n + 720^n for n >= 1. - Petros Hadjicostas, Sep 08 2019

EXAMPLE

Some solutions for n=3:

0 3 1 4 2 5 0 3 1 4 2 5 0 3 1 4 2 5 0 3 1 4 2 5

3 0 2 4 5 1 1 3 0 4 5 2 4 0 3 1 2 5 0 1 5 2 3 4

1 2 4 0 3 5 5 0 4 2 3 1 2 1 5 4 3 0 3 1 5 0 4 2

CROSSREFS

Cf. A000041, A070289, A212850, A212851, A212852, A212854, A212855, A212856, A212857, A309951, A325305.

Sequence in context: A131353 A183677 A015316 * A127660 A127659 A126165

Adjacent sequences: A212850 A212851 A212852 * A212854 A212855 A212856

KEYWORD

nonn

AUTHOR

R. H. Hardin, May 28 2012

STATUS

approved

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Last modified December 7 16:10 EST 2022. Contains 358667 sequences. (Running on oeis4.)