A268104 -id:A268104 - cp's OEIS frontend

A269435 T(n,k)=Number of length-n 0..k arrays with no repeated value greater than the previous repeated value.

2, 3, 4, 4, 9, 8, 5, 16, 27, 15, 6, 25, 64, 78, 28, 7, 36, 125, 250, 222, 51, 8, 49, 216, 615, 964, 622, 92, 9, 64, 343, 1281, 2995, 3674, 1722, 164, 10, 81, 512, 2380, 7536, 14455, 13868, 4719, 290, 11, 100, 729, 4068, 16408, 44021, 69235, 51917, 12821, 509, 12, 121

Offset: 1

R. H. Hardin, Feb 26 2016

Table starts
...2.....3......4.......5........6.........7.........8..........9.........10
...4.....9.....16......25.......36........49........64.........81........100
...8....27.....64.....125......216.......343.......512........729.......1000
..15....78....250.....615.....1281......2380......4068.......6525.......9955
..28...222....964....2995.....7536.....16408.....32152......58149......98740
..51...622...3674...14455....44021....112476....252932.....516189.....976135
..92..1722..13868...69235...255576....767172...1981512....4566213....9621220
.164..4719..51917..329430..1475871...5209554..15465934...40265487...94574110
.290.12821.192980.1558430..8482276..35236110.120310016..354051015..927338710
.509.34575.712868.7334806.48543777.237479970.933059856.3105016479.9072298237
From Robert Israel, May 30 2019: (Start)
For each of the A000110 partitions pi of the set {1,...,n}, let A_pi(n,k) be the number of length-n 0..k arrays v, such that v(i)=v(j) if and only if i and j are in the same part, and with no repeated value greater than the previous repeated value. There are restrictions on the values in the parts: if two parts a and b each have cardinality >= 2 and a_2 < b_2 (where the parts are indexed in increasing order), then v(b_i) < v(a_i). Thus if there are m partitions with cardinality >= 2, the values on those m parts are decreasing (listing these parts in order of their second entries).  So for a partition with j parts of which m have cardinality >= 2, we have A_pi(n,k) = (k+1)*k*...*(k+2-j)/m!, which is a polynomial in k of degree j.  The partition of largest cardinality is the partition into singletons, which has m=0.  The result is that for each n, T(n,k) is a monic polynomial of degree n.  To verify the "empirical" formula for a row, only n terms in that row need to be computed. (End)

			Some solutions for n=6 k=4
..1. .2. .3. .3. .2. .2. .2. .4. .3. .2. .3. .4. .3. .2. .1. .2
..4. .3. .3. .4. .0. .0. .4. .3. .1. .0. .2. .3. .4. .0. .3. .4
..3. .0. .1. .3. .3. .0. .3. .4. .3. .4. .3. .3. .3. .3. .4. .0
..3. .4. .3. .2. .1. .0. .3. .2. .1. .3. .4. .3. .2. .0. .0. .0
..3. .1. .4. .2. .2. .2. .2. .4. .3. .0. .3. .3. .3. .4. .4. .2
..2. .4. .3. .1. .3. .3. .0. .4. .2. .1. .2. .0. .0. .3. .4. .3

R. H. Hardin, Table of n, a(n) for n = 1..9999

Column 1 is A029907(n+1).
Column 2 is A268013.
Column 3 is A267975.
Diagonal is A268104.
Row 1 is A000027(n+1).
Row 2 is A000290(n+1).
Row 3 is A000578(n+1).

Empirical for column k:
k=1: a(n) = 2*a(n-1) +a(n-2) -2*a(n-3) -a(n-4)
k=2: a(n) = 6*a(n-1) -9*a(n-2) -4*a(n-3) +9*a(n-4) +6*a(n-5) +a(n-6)
k=3: [order 8]
k=4: [order 10]
k=5: [order 12]
k=6: [order 14]
k=7: [order 16]
Empirical for row n:
n=1: a(n) = n + 1
n=2: a(n) = n^2 + 2*n + 1
n=3: a(n) = n^3 + 3*n^2 + 3*n + 1
n=4: a(n) = n^4 + 4*n^3 + (11/2)*n^2 + (7/2)*n + 1
n=5: a(n) = n^5 + 5*n^4 + (17/2)*n^3 + 8*n^2 + (9/2)*n + 1
n=6: a(n) = n^6 + 6*n^5 + 12*n^4 + (44/3)*n^3 + (23/2)*n^2 + (29/6)*n + 1
n=7: a(n) = n^7 + 7*n^6 + 16*n^5 + (71/3)*n^4 + (139/6)*n^3 + (43/3)*n^2 + (35/6)*n + 1

A268246 T(n,k)=Number of nXk 0..k arrays with every repeated value in every row greater than or equal to, and every column greater than, the previous repeated value.

Original entry on oeis.org

2, 9, 4, 64, 81, 6, 615, 4096, 576, 9, 7536, 378225, 216000, 3969, 12, 112476, 56791296, 197555340, 10941048, 25281, 16, 1981512, 12650850576, 371713931400, 98956027822, 519718464, 155236, 20, 40265487, 3926389806144, 1257212773492866

Offset: 1

R. H. Hardin, Jan 29 2016

Table starts
..2......9..........64............615.............7536................112476
..4.....81........4096.........378225.........56791296...........12650850576
..6....576......216000......197555340.....371713931400......1257212773492866
..9...3969....10941048....98956027822.2335306173137957.120138605995104637373
.12..25281...519718464.46876747497685
.16.155236.23689358848
.20.915849
.25

			Some solutions for n=2 k=4
..1..1..0..1....0..0..2..0....0..1..2..0....0..2..2..2....1..1..2..3
..1..0..2..2....3..4..4..3....2..3..1..2....0..3..0..1....1..0..4..2

R. H. Hardin, Table of n, a(n) for n = 1..40

Column 1 is A002620(n+2).
Column 2 is A267961.
Column 3 is A267971.
Row 1 is A268104.
Row 2 is A268105.

Empirical for column k:
k=1: a(n) = 2*a(n-1) -2*a(n-3) +a(n-4)
k=2: [order 15]
k=3: [order 40]

cp's OEIS Frontend

A269435 T(n,k)=Number of length-n 0..k arrays with no repeated value greater than the previous repeated value.

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