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A304485
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Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.
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2
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1, 1, 0, 1, 2, 0, 1, 6, 4, 0, 1, 12, 23, 7, 0, 1, 20, 81, 73, 12, 0, 1, 30, 209, 407, 206, 19, 0, 1, 42, 451, 1566, 1751, 534, 30, 0, 1, 56, 858, 4711, 9593, 6695, 1299, 45, 0, 1, 72, 1494, 11951, 39255, 51111, 23530, 3004, 67, 0, 1, 90, 2430, 26752, 130220, 278570, 245319, 77205, 6664, 97, 0
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OFFSET
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1,5
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COMMENTS
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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.
T(n,k) is also the number of inequivalent colorings of orderless Mathematica expressions with n positions and k leaves.
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LINKS
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EXAMPLE
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Inequivalent representatives of the T(5,3) = 23 Mathematica expressions:
1[][1,1] 1[1,1][] 1[1][1] 1[1[1]] 1[1,1[]]
1[][1,2] 1[1,2][] 1[1][2] 1[1[2]] 1[1,2[]]
1[][2,2] 1[2,2][] 1[2][1] 1[2[1]] 1[2,1[]]
1[][2,3] 1[2,3][] 1[2][2] 1[2[2]] 1[2,2[]]
1[2][3] 1[2[3]] 1[2,3[]]
Triangle begins:
1
1 0
1 2 0
1 6 4 0
1 12 23 7 0
1 20 81 73 12 0
1 30 209 407 206 19 0
1 42 451 1566 1751 534 30 0
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PROG
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(PARI) \\ 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}
T(n)={my(v=Vec(InequivalentColoringsSeq(sFuncSubst(cycleIndexSeries(n), i->sv(i)*y^i)))); vector(n, n, Vecrev(v[n]/y, n))}
{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 01 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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