Sara Billey - cp's OEIS frontend

A374824 Boolean-Boolean Quilt Numbers: Triangular array T(n,k) of the number of ASM quilts of type B_n X B_k, where B_n is the Boolean lattice of subsets of an n-set ordered by inclusion.

1, 4, 16, 18, 2309, 2406862, 166, 4001278

Offset: 1

Sara Billey and Matjaz Konvalinka, Jul 21 2024

For k=1, these numbers are the Dedekind numbers given in A007153.

			Triangle begins:
    1;
    4,      16;
   18,    2309, 2406862;
  166, 4001278,     ..., ...;
  ...

Sara Billey and Matjaž Konvalinka, Generalized rank functions and quilts of alternating sign matrices, arXiv:2412.03236 [math.CO], 2024. See p. 34.

Cf. A007153, A374819, A374820, A374822, A374824.

Cf. A297622.

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.

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.

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.

Original entry on oeis.org

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.

A362554 The number of generators for the Gale submonoid of basic cyclotomic generating functions of degree n with numerator multiset bigger than denominator multiset in Gale order.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 6, 1, 5, 1, 14, 2, 9, 4, 28, 1, 33, 14, 61

Offset: 1

Sara Billey, Apr 24 2023

A basic cyclotomic generating function is a polynomial with nonnegative integer coefficients that is a product of cyclotomic polynomials, or equivalently a rational expression in q-integers indexed by disjoint numerator and denominator multisets.

S. Billey and J. Swanson, Cyclotomic Generating Functions, arXiv:2305.07620 [math.CO], 2023

A362553 Gale CGF's: The number of basic cyclotomic generating functions of degree n with numerator multiset bigger than denominator multiset in the Gale partial order.

Original entry on oeis.org

1, 1, 3, 4, 10, 12, 27, 33, 68, 82, 154, 187, 346, 410, 714, 857, 1460, 1722, 2860, 3378, 5501

Offset: 0

Sara Billey, Apr 24 2023

A basic cyclotomic generating function is a polynomial with nonnegative integer coefficients that is a product of cyclotomic polynomials, or equivalently a rational expression in q-integers indexed by disjoint numerator and denominator multisets.

S. Billey and J. Swanson, Cyclotomic Generating Functions, arXiv:2305.07620 [math.CO], 2023

A361441 The number of generators for the monoid of basic cyclotomic generating functions of degree n.

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 6, 1, 5, 3, 16, 5, 14, 6, 37, 9, 46, 33, 87

Offset: 1

Sara Billey, Mar 12 2023

A basic cyclotomic generating function is a polynomial with nonnegative integer coefficients that is a product of cyclotomic polynomials, or equivalently a rational expression in q-integers.

Sara Billey and J. Swanson, Cyclotomic Generating Functions, manuscript.

A361440 The number of generators for the monoid of basic unimodal cyclotomic generating functions of degree n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 7, 10, 9, 15, 28, 30, 34, 66, 82, 125, 126, 222, 294

Offset: 1

Sara Billey, Mar 12 2023

A basic cyclotomic generating function is a polynomial with nonnegative integer coefficients that is a product of cyclotomic polynomials, or equivalently a rational expression in q-integers.

Sara Billey and J. Swanson, Cyclotomic Generating Functions, manuscript.

A361439 The number of generators for the monoid of basic log-concave (with no internal zeros) cyclotomic generating functions of degree n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 18, 19, 37, 42, 66, 87, 132, 157, 252

Offset: 1

Sara Billey, Mar 12 2023

A cyclotomic generating function is a polynomial with nonnegative integer coefficients that is a product of cyclotomic polynomials, or equivalently a rational expression in q-integers.

Sara Billey and J. Swanson, Cyclotomic Generating Functions, manuscript.

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User: Sara Billey

A374824 Boolean-Boolean Quilt Numbers: Triangular array T(n,k) of the number of ASM quilts of type B_n X B_k, where B_n is the Boolean lattice of subsets of an n-set ordered by inclusion.

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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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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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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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A362554 The number of generators for the Gale submonoid of basic cyclotomic generating functions of degree n with numerator multiset bigger than denominator multiset in Gale order.

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A362553 Gale CGF's: The number of basic cyclotomic generating functions of degree n with numerator multiset bigger than denominator multiset in the Gale partial order.

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Keywords

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A361441 The number of generators for the monoid of basic cyclotomic generating functions of degree n.

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A361440 The number of generators for the monoid of basic unimodal cyclotomic generating functions of degree n.

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A361439 The number of generators for the monoid of basic log-concave (with no internal zeros) cyclotomic generating functions of degree n.

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