A370420 - OEIS

A370420

Number of genus g partitions of the n-set which are partitions into k nonempty subsets (blocks). Flattened 3-dimensional array read by n, then by g:0..floor(n-1)/2, then by k:1..n.

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, 1, 28, 196, 490, 490, 196, 28, 1, 0, 70, 560, 1050, 560, 70, 0, 0, 0, 28, 210, 161, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0

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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};