A304485 - OEIS

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.

%I #14 Jan 01 2021 18:12:24

%S 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,

%T 0,1,42,451,1566,1751,534,30,0,1,56,858,4711,9593,6695,1299,45,0,1,72,

%U 1494,11951,39255,51111,23530,3004,67,0,1,90,2430,26752,130220,278570,245319,77205,6664,97,0

%N 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.

%C 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.

%C T(n,k) is also the number of inequivalent colorings of orderless Mathematica expressions with n positions and k leaves.

%H Andrew Howroyd, <a href="/A304485/b304485.txt">Table of n, a(n) for n = 1..325</a> (rows 1..25)

%e Inequivalent representatives of the T(5,3) = 23 Mathematica expressions:

%e 1[][1,1] 1[1,1][] 1[1][1] 1[1[1]] 1[1,1[]]

%e 1[][1,2] 1[1,2][] 1[1][2] 1[1[2]] 1[1,2[]]

%e 1[][2,2] 1[2,2][] 1[2][1] 1[2[1]] 1[2,1[]]

%e 1[][2,3] 1[2,3][] 1[2][2] 1[2[2]] 1[2,2[]]

%e 1[2][3] 1[2[3]] 1[2,3[]]

%e Triangle begins:

%e 1

%e 1 0

%e 1 2 0

%e 1 6 4 0

%e 1 12 23 7 0

%e 1 20 81 73 12 0

%e 1 30 209 407 206 19 0

%e 1 42 451 1566 1751 534 30 0

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}

%o T(n)={my(v=Vec(InequivalentColoringsSeq(sFuncSubst(cycleIndexSeries(n), i->sv(i)*y^i)))); vector(n, n, Vecrev(v[n]/y, n))}

%o { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 01 2021

%Y Row sums are A300626.

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

%K nonn,tabl

%O 1,5

%A _Gus Wiseman_, Aug 17 2018

%E Terms a(37) and beyond from _Andrew Howroyd_, Jan 01 2021