A374514 -id:A374514 - cp's OEIS frontend

A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 11, 11, 1, 1, 1, 1, 45, 197, 45, 1, 1, 1, 1, 197, 4593, 4593, 197, 1, 1, 1, 1, 903, 126289, 732963, 126289, 903, 1, 1, 1, 1, 4279, 3888343, 155242003, 155242003, 3888343, 4279, 1, 1, 1, 1, 20793, 130016393, 40007492715, 289599115433, 40007492715, 130016393, 20793, 1, 1

Offset: 0

Andrew Howroyd, Sep 16 2024

T(n,k) is the number of normal generalized Young tableaux with all rows and columns strictly increasing whose shape is a rectangle of size n X k (cf. A299968). - Ludovic Schwob, Nov 18 2024

			Array begins:
=====================================================================
n/k | 0 1   2       3           4               5               6 ...
----+----------------------------------------------------------------
  0 | 1 1   1       1           1               1               1 ...
  1 | 1 1   1       1           1               1               1 ...
  2 | 1 1   3      11          45             197             903 ...
  3 | 1 1  11     197        4593          126289         3888343 ...
  4 | 1 1  45    4593      732963       155242003     40007492715 ...
  5 | 1 1 197  126289   155242003    289599115433 723253222084867 ...
  6 | 1 1 903 3888343 40007492715 723253222084867 ...
...
The T(2,3) = 11 matrices are:
  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 3]  [1 2 4]  [1 2 4]
  [2 3 4]  [2 4 5]  [3 4 5]  [4 5 6]  [2 3 5]  [3 4 5]
.
  [1 2 4]  [1 2 5]  [1 3 4]  [1 3 4]  [1 3 5]
  [3 5 6]  [3 4 6]  [2 4 5]  [2 5 6]  [2 4 6]

Andrew Howroyd, Table of n, a(n) for n = 0..230
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54.

Columns k=1..4 are A000012, A001003, A105124, A374985.
Main diagonal is A374514.
Cf. A060854 (case all values also distinct), A299968.

\\ See PARI link in A374514 for program code.
for(n=0, 7, print(vector(7, k, A374985(n, k-1))))

T(n,k) = T(k,n).

A376162 Number of ordered partitions of S={(i,j):1 <= i , j <= n} where for every i and j the pairs (i+1,j) and (i,j+1) are in a later part than the part containing the pair (i,j), and the pairs (i,j), (j,i) are in the same part.

Original entry on oeis.org

1, 1, 3, 39, 2905, 1538369, 6904262355, 304662492057063, 150347237334006997801, 929721796071361437087789041, 79773595676787229793797978773561927, 104165556509336140832819242491033872033130063, 2252283824141388832759484222915451435885285752729087857

Offset: 1

Kevin O'Bryant, Sep 12 2024

nonn

Ordered partitions are also called weak orderings.

Any such ordered partition can be written as a list of pairs (i,j) with 1 <= i <= j <= n, with either "=" or "<" between each pair, and so that (i,j) appears in the list before (i+1,j) (if i

Given any set A={a_1<...

Equivalently, a(n) is the number of n X n symmetric matrices whose values cover an initial interval of positive integers and whose rows have values which are strictly increasing. - Andrew Howroyd, Sep 15 2024

			For n=2 the a(2)=1 ordered partition is {(1,1)}<{(2,1),(1,2)}<{(2,2)}. We can encode this as 11<12<22, writing "ij" for the pair (i,j).
For n=3 one of the a(3)=3 ordered partitions is {(1,1)}<{(1,2),(2,1)}<{(1,3),(3,1),(2,2)}<{(2,3),(3,2)}<{(3,3)}, which is encoded as either 11<12<13=22<23<33 or 11<12<22=13<23<33. The other two ordered partitions can be encoded as 11<12<22<13<23<33 and 11<12<13<22<23<33.
From _Andrew Howroyd_, Sep 15 2024: (Start)
The a(3) = 3 symmetric matrices are:
    [1 2 3]   [1 2 3]   [1 2 4]
    [2 3 4]   [2 4 5]   [2 3 5]
    [3 4 5]   [3 5 6]   [4 5 6]
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..20

Cf. A000670, A006541, A003121, A374514.

\\ See PARI link in A374514 for program code.
vector(10, n, A376162(n)) \\ Andrew Howroyd, Sep 16 2024

a(n) <= A000670(n*(n+1)/2).

a(7) onwards from Andrew Howroyd, Sep 15 2024

A376504 Number of divisors of n that are both composite and squarefree.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 4, 0, 1, 1, 0, 1, 4, 0, 1, 1, 4, 0, 1, 0, 1, 1, 1, 1, 4, 0, 1, 0, 1, 0, 4, 1, 1, 1

Offset: 1

Michael De Vlieger, Sep 25 2024

Also number of composite and squarefree m <= n such that rad(m) | n, i.e., in row n of A162306, where rad = A007947.
This sequence is distinct from A327517; A327517(210) != a(210).
Record setters are primorials, a(6) = 1, a(30) = 4, a(210) = 11, etc., since primorials P(n) = A002110(n) are the smallest instance of omega(n) = A001221(n).

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Michael De Vlieger, Hasse diagram of row 1440 of A162306 showing 4 squarefree composites in green, 3 primes in red, the empty product in gray, 17 perfect powers of primes in yellow, and 72 numbers that are neither squarefree nor prime powers in blue and purple, with purple additionally representing powerful numbers that are not prime powers.

Cf. A000005, A000295, A000961, A001221, A002110, A007947, A034444, A120944, A162306, A327517, A361373 (number of prime powers in row n of A162306), A374514 (number of divisors of n that are neither squarefree nor prime powers).

Mathematica
```
Array[2^# - # - 1 &@ PrimeNu[#] &, 120]
```

a(n) = 2^omega(n) - omega(n) - 1 = A034444(n) - A001221(n) - 1.
a(n) = 0 for n = p^m, where p is prime and m >= 0, i.e., n in A000961.
a(n) = A000295(omega(n)) = A000295(A001221(n)).

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A374985 Array read by antidiagonals: T(n,k) is the number of n X k matrices whose values cover an initial interval of positive integers and whose rows and columns have values which are strictly increasing.

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A376162 Number of ordered partitions of S={(i,j):1 <= i , j <= n} where for every i and j the pairs (i+1,j) and (i,j+1) are in a later part than the part containing the pair (i,j), and the pairs (i,j), (j,i) are in the same part.

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A376504 Number of divisors of n that are both composite and squarefree.

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Author

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Links

Crossrefs

Programs

Mathematica

Formula