Matjaz Konvalinka - 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.

A258620 Number of tanglegrams of size n.

Original entry on oeis.org

1, 1, 2, 13, 114, 1509, 25595, 535753, 13305590, 382728552, 12515198465, 458621603279, 18619063906689, 829607273337513, 40253392454978755, 2112878091130119496, 119296114546292088543, 7209829960147215492897, 464413707136960430809460, 31762965767675300603026848

Offset: 1

Matjaz Konvalinka, Jun 18 2015

nonn

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Matjaz Konvalinka, Table of n, a(n) for n = 1..366
S. C. Billey, M. Konvalinka, and F. A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.
Sara Billey, Matjaž Konvalinka, Frederick A. Matsen IV, On trees, tanglegrams, and tangled chains, hal-02173394 [math.CO], 2020.
M. Konvalinka, S. Wagner, The shape of random tanglegrams, arXiv preprint arXiv:1512.01168, 2015.
Dimbinaina Ralaivaosaona, Jean Bernoulli Ravelomanana, Stephan Wagner, Counting Planar Tanglegrams, LIPIcs Proceedings of Analysis of Algorithms 2018, Vol. 110. Article 32.

r[h_, n_, s_] :=
  r[h, n, s] =
   If[n == 0, 1,
    Sum[Product[(2 (s + j 2^h) - 1)^2/(j 2^h), {j, m}] r[
       h + 1, (n - m)/2, s + m 2^h], {m, n, 0, -2}]];
tang[n_] := r[0, n, 0]/(2 n - 1)^2;

a(n) = Sum_{lambda binary partition of n} (Product_{i=2..l(lambda)} (2(lambda_i+...+lambda_l)-1)^2)/z_lambda.

a(n) ~ 2^(2*n-3/2) * n^(n-5/2) / (sqrt(Pi) * exp(n-1/8)).

A258489 Number of tangled chains of length k=6.

Original entry on oeis.org

1, 1, 122, 474883, 11168414844, 989169269347359, 250335000079534559375, 151038989624520433840089358, 191158216491241179675824199407135, 461408865973380293005829125668717407727, 1973397409908124305318632313047269426852165625, 14104214451439837037643144221899175649593123932192274

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=6, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^6)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A258488 Number of tangled chains of length k=5.

Original entry on oeis.org

1, 1, 41, 31732, 106420469, 1046976648840, 24085106680575625, 1117767454807330938472, 94308987414050519542935029, 13390317159105772877158700776107, 3014130596940522685213135526859317500, 1025828273466214412416440210115479183065903, 507888918625036626314714587415852381698509422634

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=5, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^5)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A258485 Number of tangled chains of length k=7.

Original entry on oeis.org

1, 1, 365, 7119961, 1172597933594, 934741501255380321, 2602204282373953017437500, 20410544568790568555722851029455, 387481340785957748099474582410763014214, 15899856312608503503306403988460714538830399657

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=6, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^7)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

A258487 Number of tangled chains of length k=4.

Original entry on oeis.org

1, 1, 14, 2140, 1017219, 1110178602, 2320017306125, 8278981347401059, 46556715158334549170, 388779284837787599307987, 4605471565794120802036550000, 74633554055057890778698344509705, 1606481673354648219373898238155693682, 44821655543075499856527523557216582931002

Offset: 1

Sara Billey, Matjaz Konvalinka, and Frederick A. Matsen IV, May 31 2015

nonn

Tangled chains are ordered lists of k rooted binary trees with n leaves and a matching between each leaf from the i-th tree with a unique leaf from the (i+1)-st tree up to isomorphism on the binary trees. This sequence fixes k=4, and n = 1,2,3,...

R. Page, Tangled trees: phylogeny, cospeciation, and coevolution, The University of Chicago Press, 2002.

S. Billey, Table of n, a(n) for n = 1..20
Sara Billey, Matjaž Konvalinka, and Frederick A. Matsen IV, On the enumeration of tanglegrams and tangled chains, arXiv:1507.04976 [math.CO], 2015.

Cf. A000123 (binary partitions), A258620 (tanglegrams), A258485, A258486, A258487, A258488, A258489 (tangled chains), A259114 (unordered tanglegrams).

t(n) = Sum_{b=(b(1),...,b(t))} Product_{i=2..t} (2(b(i)+...+b(t))-1)^4)/z(b) where the sum is over all binary partitions of n and z(b) is the size of the stabilizer of a permutation of cycle type b under conjugation.

cp's OEIS Frontend

User: Matjaz Konvalinka

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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A258620 Number of tanglegrams of size n.

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A258489 Number of tangled chains of length k=6.

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A258488 Number of tangled chains of length k=5.

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A258485 Number of tangled chains of length k=7.

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A258487 Number of tangled chains of length k=4.

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