A374824 -id:A374824 - cp's OEIS frontend

A374819 Triangle read by rows: T(n,k) is the number of functions on the Boolean lattice B_n satisfying f({}) =0, f([n])=k, and the Boolean growth rule: f(J union {i})-f(J) in {0,1} for all subsets J of [n]={1, ..., n} and all i in [n]\J, 0 <= k <= n.

1, 1, 1, 1, 4, 1, 1, 18, 18, 1, 1, 166, 656, 166, 1, 1, 7579, 189967, 189967, 7579, 1, 1, 7828352

Offset: 0

Sara Billey and Matjaz Konvalinka, Jul 25 2024

For k=1, these numbers are the Dedekind numbers A007153 counting the number of monotone Boolean functions or equivalently antichains of subsets of an n-set containing at least one nonempty set.

			Triangle begins:
  1;
  1,    1;
  1,    4,      1;
  1,   18,     18,      1;
  1,  166,    656,    166,    1;
  1, 7579, 189967, 189967, 7579, 1;
  ...

Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 32.

Cf. A000372, A007153, A374820, A374821, A374822, A297622, A374824.

A374820 Boolean-Chain Quilt Numbers: Square table of the number of ASM quilts of type B_n X C_k read down antidiagonals, where B_n is the Boolean lattice on an n-set and C_k is a chain of length k with k+1 elements.

Original entry on oeis.org

1, 2, 4, 3, 4, 18, 4, 17, 199, 166, 5, 46, 199, 47000, 7579, 6, 100, 3252, 3813042, 410131245, 7828352

Offset: 1

Sara Billey and Matjaz Konvalinka, Jul 21 2024

For k=1, these numbers are the Dedekind numbers A007153 counting the number of monotone Boolean functions or antichains of subsets of an n-set containing at least one nonempty set.
For k=2, these numbers are Antichain-Boolean numbers, see A374821.
These numbers are given by a polynomial in k for fixed n when k>=n.

			Square array begins:
        1,         2,       3,    4,   5,  6, ...
        4,         4,      17,   46, 100, ...
       18,       199,     199, 3252, ...
      166,     47000, 3813042, ...
     7579, 410131245, ...
  7828352, ...

Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 32.

Cf. A374819, A374821, A374822, A297622, A374824.

A374822 Antichain-Chain Quilt Numbers: Square table of the number of ASM quilts of type A_2(j) x C_k read down antidiagonals, where C_k is the chain poset or rank k and A_2(j) is the rank 2 poset with a unique minimal and maximal element and j atoms.

Original entry on oeis.org

2, 4, 2, 8, 4, 7, 16, 8, 17, 16, 32, 16, 43, 46, 30, 64, 32, 113, 142, 100, 50, 128, 64, 307, 466, 366, 190, 77, 256, 128, 857, 1606, 1444, 806, 329, 112, 512, 256, 2443, 5746, 6030, 3718, 1589, 532, 156, 1024, 512, 7073, 21142, 26260, 18230, 8393, 2884, 816

Offset: 1

Sara Billey and Matjaz Konvalinka, Jul 21 2024

			Array begins:
  2,  4,  8, ...
  2,  4,  8, ...
  7, 17, 43, ...
  ...

Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.

Cf. A374819, A374820, A374821, A297622, A374824.

T(j,k) = Sum_{i=2..k} (k+1-i)*i^j for j>=1 and k>1.

T(j,1) = 2^j for all j>=1 and k=1.

A374821 Antichain-Boolean Quilt Numbers: Square table of the number of ASM quilts of type B_n x A_2(j) read down antidiagonals, where B_n is the Boolean lattice and A_2(j) is the rank 2 poset with a unique minimal and maximal element and j atoms.

Original entry on oeis.org

2, 4, 4, 8, 16, 199, 16, 64, 2309, 47000, 32, 256, 28225, 4001278, 410131245, 64, 1024, 364217, 384285926

Offset: 1

Sara Billey and Matjaz Konvalinka, Jul 21 2024

Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 33.

Cf. A374819, A374820, A374822, A297622, A374824.

cp's OEIS Frontend

A374819 Triangle read by rows: T(n,k) is the number of functions on the Boolean lattice B_n satisfying f({}) =0, f([n])=k, and the Boolean growth rule: f(J union {i})-f(J) in {0,1} for all subsets J of [n]={1, ..., n} and all i in [n]\J, 0 <= k <= n.

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A374820 Boolean-Chain Quilt Numbers: Square table of the number of ASM quilts of type B_n X C_k read down antidiagonals, where B_n is the Boolean lattice on an n-set and C_k is a chain of length k with k+1 elements.

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A374822 Antichain-Chain Quilt Numbers: Square table of the number of ASM quilts of type A_2(j) x C_k read down antidiagonals, where C_k is the chain poset or rank k and A_2(j) is the rank 2 poset with a unique minimal and maximal element and j atoms.

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A374821 Antichain-Boolean Quilt Numbers: Square table of the number of ASM quilts of type B_n x A_2(j) read down antidiagonals, where B_n is the Boolean lattice and A_2(j) is the rank 2 poset with a unique minimal and maximal element and j atoms.

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