OFFSET
1,5
COMMENTS
Genus-dependent Stirling numbers of the second kind S2(n,k,g), 1 <= n, 1 <= k <= n, 0 <= g <= floor((n-1)/2). This is an infinite three-dimensional array. Its first 15 rows (n:1..15) are given by the table (see Links) taken from the article by Robert Coquereaux and Jean-Bernard Zuber (where a transpose of this table is given), see p. 32. These 15 rows determine 589 entries of the sequence (Data).
Example: the numbers S2(5,k,0), k=1..5, are {1,10,20,10,1} and appear on line 5, column 1; the numbers S2(5,k,1), k=1..5, are {0,5,5,0,0} and appear on line 5, column 2. Values of S2(n,k,g) for g > floor((n-1)/2) are equal to 0 and are not displayed.
Summing S2(n,k,g) over k gives genus-dependent Bell numbers B(n,g), A370235. Summing S2(n,k,g) over g gives S2(n,k), the Stirling numbers of the second kind A008277. Summing S2(n,k,g) over k and g gives the Bell numbers B(n), A000110. Example: S2(5,k,0) = 1, 10, 20, 10, 1 and S2(5,k,1) = 0, 5, 5, 0, 0 for k = 1..5; therefore S2(5,k) = 1, 15, 25, 10, 1, B(5,0) = 42, B(5,1) = 10, and B(5) = 52.
LINKS
Robert Coquereaux, Rows n = 1..15 of array, flattened, terms 1..589
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 4, 5, 22.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus: a compendium of results, Journal of Integer Sequences, Vol. 27 (2024), Article 24.2.6. See p. 8, 9, 10, 32.
Robert Coquereaux, Table of S2(n,k,g) for n = 1..15
Robert Coquereaux, Mathematica program for the numbers S(n,k,g) and B(n,g)
Robert Cori and Gabor Hetyei, Counting partitions of a fixed genus, Electron. J. Combin. 25 (4) (2018), #P4.26.
Jean-Bernard Zuber, Counting partitions by genus. I. Genus 0 to 2, arXiv:2303.05875 [math.CO], 2023.
Jean-Bernard Zuber, Counting partitions by genus. I. Genus 0 to 2, Enumer. Comb. Appl. 4 (2) (2024) #S2R13.
FORMULA
No general formula is currently known. In the particular cases g=0, 1, 2, a formula is known: see Crossrefs.
EXAMPLE
For n:1..7, g:1..floor(n-1)/2, k:1..n. The 3-dimensional array begins:
{1};
{1,1};
{1,3,1};
{1,6,6,1}, {0,1,0,0};
{1,10,20,10,1}, {0,5,5,0,0};
{1,15,50,50,15,1}, {0,15,40,15,0,0}, {0,1,0,0,0,0};
{1,21,105,175,105,21,1}, {0,35,175,175,35,0,0}, {0,7,21,0,0,0,0};
MATHEMATICA
See Links
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Robert Coquereaux, Feb 18 2024
STATUS
approved