A304485 -id:A304485 - cp's OEIS frontend

A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

1, 1, 3, 11, 43, 187, 872, 4375, 23258, 130485, 767348, 4710715, 30070205, 198983975, 1361361925, 9607908808, 69812787049, 521377973359, 3996036977270, 31389624598631, 252408597286705, 2075472033455894, 17434190966525003, 149476993511444023, 1307022313790487959

Offset: 0

Gus Wiseman, Aug 17 2018

nonn

A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.

Also the number of inequivalent colorings of orderless Mathematica expressions with n positions.

			Inequivalent representatives of the a(3) = 11 colorings:
  1[1,1]  1[2,2]  1[1,2]  1[2,3]
  1[1[]]  1[2[]]
  1[][1]  1[][2]
  1[1][]  1[2][]
  1[][][]

Row sums of A304485.

Cf. A000612, A007716, A052893, A053492, A277996, A279944, A280000, A317652, A317655, A317656, A317676.

\\ See links in A339645 for combinatorial species functions.
cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
InequivalentColoringsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

cp's OEIS Frontend

A300626 Number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions.

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