A304485 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.

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

Offset: 1

Author

Gus Wiseman, Aug 17 2018

Keywords

• nonn
• tabl

Comments

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.

Examples

			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
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..325 (rows 1..25)

Crossrefs

Row sums are A300626.

Cf. A000612, A007716, A052893, A053492, A277996, A280000, A317676.

Programs

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

Extensions

Terms a(37) and beyond from Andrew Howroyd, Jan 01 2021