cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A297178 Triangle read by rows: T(n,k) = number of partitions of genus 2 of n elements with k parts (n >= 6, 2 <= k <= n-4).

Original entry on oeis.org

1, 7, 21, 28, 210, 161, 84, 1134, 2184, 777, 210, 4410, 15330, 13713, 2835, 462, 13860, 75075, 121275, 63063, 8547, 924, 37422, 289905, 729960, 685608, 233772, 22407, 1716, 90090, 942942, 3396393, 4972968, 3063060, 738738, 52767, 3003, 198198, 2690688, 13096083, 27432405, 26342316, 11477466, 2063061, 114114
Offset: 6

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Author

N. J. A. Sloane, Dec 26 2017

Keywords

Examples

			Triangle begins (see Table 3.2 in Yip's thesis):
    1;
    7,    21;
   28,   210,    161;
   84,  1134,   2184,    777;
  210,  4410,  15330,  13713,   2835;
  462, 13860,  75075, 121275,  63063,   8547;
  924, 37422, 289905, 729960, 685608, 233772, 22407;
  ...

Links

Crossrefs

Row sums are A297179.
First column is A000579.

Programs

  • Mathematica
    T[n_,k_]:=((-6*(-2 + n)*(-1 + n) - k^2*(-13 + 5*n) + k*(-8 + n*(-9 + 5*n)))*(-4 + n)!*n!)/(1440*(-2 + k)!*k!*(-4 - k + n)!*(-k + n)!) (* Robert Coquereaux, Mar 05 2024 *)

Formula

T(n,k) = 8*gam(n-10,k-6) -4*gam(n-10,k-5) -15*gam(n-10,k-4) +10*gam(n-10,k-3) +gam(n-10,k-2) -4*gam(n-9,k-5) +39*gam(n-9,k-4) -10*gam(n-9,k-3) -4*gam(n-9,k-2) -15*gam(n-8,k-4) -10*gam(n-8,k-3) +6*gam(n-8,k-2) -4*gam(n-7,k-2) +10*gam(n-7,k-3) +gam(n-6,k-2) with gam(n,k) = (binomial(n+10,5) * binomial(n+5,k) * binomial(n+5,n-k)) / binomial(10,5) [Cori & Hetyei]. - Robert Coquereaux, Feb 12 2024
T(n,k) = ((-6*(-2 + n)*(-1 + n) - k^2*(-13 + 5*n) + k*(-8 + n*(-9 + 5*n)))*(-4 + n)!*n!) / (1440*(-2 + k)!*k!*(-4 - k + n)!*(-k + n)!). - Robert Coquereaux, Mar 05 2024

A370235 Table read by rows. Number of set partitions of [n] with respect to genus g.

Original entry on oeis.org

1, 1, 2, 5, 14, 1, 42, 10, 132, 70, 1, 429, 420, 28, 1430, 2310, 399, 1, 4862, 12012, 4179, 94, 16796, 60060, 36498, 2620, 1, 58786, 291720, 282282, 45430, 352, 208012, 1385670, 1999998, 600655, 19261, 1, 742900, 6466460, 13258674, 6633484, 541541, 1378
Offset: 0

Views

Author

Peter Luschny, Feb 15 2024

Keywords

Comments

The table shows the number of partitions of [n] = {1, 2, 3, ..., n} with genus g.
The set of noncrossing partitions is exactly the set of genus zero partitions. The numbers corresponding to this case are the Catalan numbers.
This is essentially table 2.1 in Martha Yip's thesis (p. 12).
From Robert Coquereaux, Feb 16 2024: (Start)
The two-dimensional array is called triangle of genus-dependent Bell numbers B(n, g); if n >= 1, n even, nonzero values are obtained for 0 <= g <= floor((n-1)/2); if n >= 1, odd, nonzero values are obtained for 0 <= g < (n-1)/2.
The two-dimensional array B(n, g) can be obtained from a three-dimensional array S2(n, k, g), by summation over the number k of blocks. The numbers S2(n, k, g) are genus-dependent Stirling numbers of the second kind. They give the number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). The numbers S2(n, k, g) are discussed in A370420.
(End)

Examples

			[n\g]     0        1        2      3      4     5
-------------------------------------------------
[ 0]      1;
[ 1]      1;
[ 2]      2;
[ 3]      5;
[ 4]     14,       1;
[ 5]     42,      10;
[ 6]    132,      70,        1;
[ 7]    429,     420,       28;
[ 8]   1430,    2310,      399,       1;
[ 9]   4862,   12012,     4179,      94;
[10]  16796,   60060,    36498,    2620,      1;
[11]  58786,  291720,   282282,   45430,    352;
[12] 208012, 1385670,  1999998,  600655,  19261,    1;
[13] 742900, 6466460, 13258674, 6633484, 541541, 1378;

Links

Crossrefs

Columns: A000108 (g=0), A002802 (g=1), A297179 (g=2), A370237 (g=3).
Cf. A000110 (row sums), A177267 (permutations by genus).
Cf. A001263, A370236, A297178.
Cf. A370420 (S2(n,k,g)).

A370237 Number of genus 3 partitions of the n-set.

Original entry on oeis.org

1, 94, 2620, 45430, 600655, 6633484, 64336844, 565256120
Offset: 8

Views

Author

Robert Coquereaux, Feb 12 2024

Keywords

Comments

Call B(n, g) the number of genus g partitions of a set with n elements (genus-dependent Bell number). Then a(n) = B(n, 3) with B(8, 3) = 1.
a(8) = 1 through a(15) = 565256120 were explicitly determined by listing of partitions of an n-set and selecting those of genus 3.
The coefficients of the sixth-degree polynomial appearing in the numerator of the conjectured formula were determined by using experimental values for a(8) up to a(14); the term a(15) given by the formula agrees with the experimental value.
Using the conjectured formula for a(n) gives the following terms for n=16..20 : 4593034160, 35025118700, 253374008888, 1753071498620, 11675101781850. The E.g.f. given in the Formula section is obtained from the conjectured formula for a(n).

Links

Crossrefs

Column g=3 of A370235.
Cf. A000108, A002802, A297179, A001263, A370236, A297178.

Formula

Conjecture: a(n) = (1/(2^13 * 3^4 * 5 * 7)) * (35*n^6 - 819*n^5 + 7589*n^4 - 36009*n^3 + 93464*n^2 - 129060*n + 95040)/((2*n - 11)*(2*n - 9)*(2*n - 7)*(2*n - 5)*(2*n - 3)*(2*n - 1)) * (1/(n-8)!) * (2*n)!/n!.
Conjecture: E.g.f.: (1/181440)*exp(2*x)*(x^2*(720 - 720*x + 1080*x^2 - 720*x^3 + 537*x^4 - 294*x^5 + 140*x^6)*BesselI(0, 2*x) + x*(-720 + 720*x - 1440*x^2 + 1080*x^3 - 1017*x^4 + 594*x^5 - 329*x^6 + 140*x^7)*BesselI(1, 2*x)).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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