A306372 -id:A306372 - cp's OEIS frontend

A323849 Irregular triangle read by rows: T(n,d) (n >= 1, 0 <= d <= 2n-2) = number of n X n integer-valued matrices M such that M_{1,1}=0, M_{n,n}=d, and M_{(i+1),j} = M_{i,j} + (0 or 1), M_{i,(j+1)} = M_{i,j} + (0 or 1).

1, 1, 4, 1, 1, 18, 44, 18, 1, 1, 68, 615, 1236, 615, 68, 1, 1, 250, 7313, 46812, 84910, 46812, 7313, 250, 1, 1, 922, 85801, 1592348, 8241540, 14024408, 8241540, 1592348, 85801, 922, 1, 1, 3430, 1030330, 54926890, 759337545, 3397542544, 5530983756, 3397542544, 759337545, 54926890, 1030330, 3430, 1

Offset: 1

N. J. A. Sloane, Feb 07 2019

			Triangle begins:
  n\d 0   1     2       3       4        5       6       7     8   9 10
  1   1
  2   1   4     1
  3   1  18    44      18       1
  4   1  68   615    1236     615       68       1
  5   1 250  7313   46812   84910    46812    7313     250     1
  6   1 922 85801 1592348 8241540 14024408 8241540 1592348 85801 922  1
  ...

D. E. Knuth, Email to N. J. A. Sloane, Feb 06 2019.

Alois P. Heinz, Rows n = 1..15, flattened

Columns k=0-2 give: A000012, A115112, A252869.
T(n,n-1) gives A306372.
Cf. A323848.

T(n,1) = binomial(2n,n) - 2 = A115112(n).

The triangle is symmetric: T(n,d) = T(n,2n-2-d).

Edited by Alois P. Heinz, Feb 11 2019

A367432 Number of commutative discrete aggregation functions defined on the finite chain L_n={0,1,...,n-1,n} that are smooth.

Original entry on oeis.org

1, 2, 10, 80, 1008, 19764, 600028, 28134464, 2034669118, 226781039624

Offset: 0

Marc Munar, Nov 18 2023

The number of smooth and commutative discrete aggregation functions on the finite chain L_n={0,1,...,n-1,n}, i.e., the number of monotonic increasing binary functions F: L_n^2->L_n such that F(0,0)=0 and F(n,n)=n, F(x,y)=F(y,x) for all x,y in L_n (commutativity), and F(x+1,y)-F(x,y)<=1 and F(y,x+1)-F(y,x)<=1 for all y in L_n and x in L_n\{n} (smooth).

Also, the number of (n+1)X(n+1) integer symmetric matrices (m_{i,j}) such that m_{1,1}=1, m_{n+1,n+1}=n+1, and all rows and columns are (weakly) monotonic without jumps larger than 1.

Symmetric counterpart of matrices enumerated in A306372.

Smooth counterpart of operators defined in A366447.

a(0) and a(7)-a(9) from Martin Ehrenstein, Dec 01 2023

cp's OEIS Frontend

A323849 Irregular triangle read by rows: T(n,d) (n >= 1, 0 <= d <= 2n-2) = number of n X n integer-valued matrices M such that M_{1,1}=0, M_{n,n}=d, and M_{(i+1),j} = M_{i,j} + (0 or 1), M_{i,(j+1)} = M_{i,j} + (0 or 1).

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