A267975 -id:A267975 - 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

A267997 T(n,k) = number of n X k 0..3 arrays with every repeated value in every row and column greater than or equal to the previous repeated value.

Original entry on oeis.org

4, 16, 16, 64, 256, 64, 250, 4096, 4096, 250, 964, 62500, 262144, 62500, 964, 3674, 929296, 15625000, 15625000, 929296, 3674, 13868, 13498276, 895841344, 3552627426, 895841344, 13498276, 13868, 51917, 192321424, 49592666024, 766970396055, 766970396055, 49592666024, 192321424, 51917

Offset: 1

R. H. Hardin, Jan 24 2016

			Table starts:
       4          16              64               250                964
      16         256            4096             62500             929296
      64        4096          262144          15625000          895841344
     250       62500        15625000        3552627426       766970396055
     964      929296       895841344      766970396055    615504325372174
    3674    13498276     49592666024   158016399692284 465926430628469626
   13868   192321424   2667113508032 31323853192497584 ...
   51917  2695374889 139935778112213 ...
  192980 37241280400 ...
  712868 ...
  ...
Some solutions for n=2 k=4:
..3..1..2..0....0..1..3..3....1..2..1..1....3..2..3..1....2..2..0..1
..3..0..3..0....1..0..2..0....0..3..3..3....3..2..0..3....2..1..0..2

R. H. Hardin, Table of n, a(n) for n = 1..60

Columns 1..5 are A267975, A267976, A267994, A267995, A267996.

Empirical for column k:
k=1: [linear recurrence of order 8]
k=2: [order 21]
k=3: [order 40]

Name corrected by Andrew Howroyd, Mar 21 2025

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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