Walks, Partitions, and Normal Ordering

We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the n-th Weyl algebra. Under several specifications , we study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the λ-Stirling numbers.


Introduction
Let G = (V, E) be a digraph with an ordered set of edges E = (e 1 , . . ., e m ).A walk of G is any sequence of edges e i 1 . . .e i such that the terminal vertex of e i k coincides with the initial vertex of e i k+1 for all k < .The walk e i 1 . . .e i is called Consider decompositions of G into edge-disjoint increasing walks.This setting generalizes set partitions, because when G has only one vertex and m labeled loops (1,1), decompositions into increasing walks correspond to partitions of the set [m] := {1, . . ., m} into subsets.If G is decomposed into one walk (with distinct edges), it is an Euler tour.If G is a path and edges along the path are labeled by an arbitrary permutation σ ∈ S m , decompositions into minimal number of increasing walks index the descent set of σ.
This interpretation arises from the normal ordering problem in the Weyl algebra.The n-th Weyl algebra A n is an associative algebra with 2n generators x 1 , . . ., x n , ∂ 1 , . . ., ∂ n subject to relations [∂ i , x j ] = δ i,j , [x i , x j ] = 0, [∂ i , ∂ j ] = 0, where [a, b] = ab − ba is the commutator and δ i,j is the Kronecker delta.The element w ∈ A n is normally ordered if it is expressed in the form w = k c k i x i j ∂ j .For the digraph G, the special case of our normal ordering formula is where (i , j ) is an -th edge of G and J is the corresponding multiset of sinks (which is determined uniquely from the given sources I).For example, the graph in Figure 1 has four decompositions into increasing walks: e 1 e 2 e 3 , e 1 e 2 ∪ e 3 , e 1 ∪ e 2 e 3 , e 1 ∪ e 2 ∪ e 3 .
This corresponds to the normal ordering where the sources and sinks of walks are exactly the indices of terms.In fact, this formula is a graph-theoretic version of Olshanski's analog of Wick's formula [26] (cf.[27]).Collecting the terms we can rewrite where S G (I) is the number of decompositions of G into increasing walks with multiset of sources I.In the case of the first Weyl algebra A 1 this formula gives the well-known expansion (x∂) m = i S(m, i)x i ∂ i for Stirling numbers of the second kind S(m, i).
We introduced this combinatorial model in a more general setting in [15], where we studied algebraic applications, polynomial identities and commutators on Weyl algebra.In this paper we focus on combinatorial aspects of walk decompositions and various specifications of the normal ordering interpretations, such as restricted set partitions and generalized Stirling numbers.We study walk decompositions and the G-Stirling functions which enumerate decompositions by sources of walks.As we see, the values of the G-Stirling function serve as connection constants in the normal ordering problem.
The setting of decomposing graphs into walks is a source for certain types of restricted set partitions.For example, assume that digraph has 2 vertices and edges e 1 , e 3 , . . .going from vertex 1 to 2 and the remaining edges e 2 , e 4 , . . .going back from 2 to 1. Decompositions on this model generate set partitions with the property that each block (when sorted) is parity alternating (i.e.odd, even, odd, etc.).We show that the total number of parity alternating partitions of the set [m], satisfies the formula a(m) = B (m+1)/2 B (m+1)/2 , where B k is the k-th Bell number, the number of partitions of [k].Note that the latter formula is not an obvious fact from the definition of a(m).We apply the composition of operators in Weyl algebra to compute the number of such decompositions.This approach is illustrated for a new special type of set partitions, which we call the residue alternating partitions (a general version of the parity alternating partitions).The elements in every block of these partitions form a consecutive (cyclic) interval modulo n and their total number is the product of Bell numbers.We show both algebraic and bijective proofs to this fact.
We then specify our interpretations to the case n = 1, for which there are many related studies, e.g.[3,4,12,18,20,21,24,30] and refer to a recent book [22] on the subject of normal ordering.We introduce and study the λ-Stirling numbers.These numbers of second kind naturally appear in decompositions of graphs with one vertex and many loops.For the given sequence λ = (λ 1 λ 2 • • • ), the λ-Stirling numbers of the second kind S λ (n, k) is the number of partitions of [n] into k blocks such that the first λ 1 elements of [n] are in distinct blocks, the next λ 2 elements are in distinct blocks, and so on.This definition is a natural generalization of the r-Stirling numbers [5] and it was studied in [25].We show how these numbers S λ (n, k) arise from our general graph setting as well as from differential operators in the first Weyl algebra.On the other hand, we also define the corresponding dual λ-Stirling numbers of the first kind C λ (n, k), and show a combinatorial interpretation to them.Namely, C λ (n, k) is the number of permutations of [n] having k cycles such that non-minimal elements of the first λ 1 cycles are greater than all minimal elements of these λ 1 cycles; non-minimal elements of the next λ 2 cycles are greater than all minimal elements of these λ 2 cycles, and so on (some of the remaining cycles are singletons).The classical Stirling numbers are defined on two parameters: the number of elements and the number of blocks (cycles).For these generalizations, one can see that the λ-sequence affects on the first parameter in λ-Stirling numbers of the second kind, and affects on the second parameter (cycles) in λ-Stirling numbers of the first kind.We obtain many properties of S λ (n, k), C λ (n, k) analogous to the properties of the usual Stirling numbers.

Decompositions into increasing walks
Suppose that edges of the digraph G = (V, E) are ordered (or labeled), E = (e 1 , . . ., e m ).A k-decomposition is a decompositions of E into k edge-disjoint walks.We say that the k-decomposition E = P 1 ∪ . . .∪ P k is principal if every walk P i = e 1 . . .e s (1 i k) is increasing, i.e. we have 1 < . . .< s .
When V = {1} and graph has m labeled loop edges (1, 1), principal decompositions correspond to partitions of the set [m] into disjoint subsets.Further, we suppose that the digraph G is presented by the vertex set V = [n].
A block (or p-block if p is specified) is a distinguished set of edges {e 1 , . . ., e p }.If graph is built up from several (disjoint) blocks, then we will require that the edges in each block must lie in distinct walks.For example, digraph in Figure 2 which is built from three blocks B 1 = {e 1 }, B 2 = {e 2 , e 3 }, B 3 = {e 4 , e 5 , e 6 }, has a principal 4-decomposition e 1 e 5 ∪ e 2 ∪ e 3 e 4 ∪ e 6 .Note that decomposition e 1 e 5 ∪ e 2 e 3 e 4 ∪ e 6 cannot be used here since e 2 , e 3 are from the same block B 1 and thus cannot be in the same path.
For each vertex v ∈ V consider the sets of its incoming and outcoming edges, respectively (loops (v, v) are included in both sets).Suppose that the edges of G are built up from some partition of E into blocks.Consider the matchings between In(v) and Out(v) defined as follows.We allow elements e i ∈ In(v) and e j ∈ Out(v) to be matched if i < j and e i , e j are not in the same block.A matching now is defined as some set of such matched pairs, where every edge is used at most once from In(v) and at most once from Out(v) (so, only loops can be used once for both In(v), Out(v)).Let M(v) be the set of all possible matchings (not necessarily maximal, and including an empty matching) and M (v) be the set of maximal matchings.For example, at vertex 2 of the graph in Figure 2  Let PD(G) be the set of principal decompositions of G.
Proof.Let us take an arbitrary matching for every vertex v (1 v n) and construct a principal graph decomposition.If e i ∈ In(v) and e j ∈ Out(v) are matched, then we define them to be a fragment of a path e i e j .Otherwise, if e i ∈ In(v) or e j ∈ Out(v) are unmatched edges, then define them as final and initial edges of their corresponding paths, respectively.One can easily verify that this map defines a principal decomposition and its inverse defines a matching for every vertex.
the electronic journal of combinatorics 22(4) (2015), #P4.10 Corollary 2.3.If G is acyclic digraph, then there exists a labeling of its edges such that every path is increasing.Therefore, every decomposition of G is a principal decomposition for such labeling.For every vertex v we can match some i edges from In(v) with some i edges of In(v) in in(v) i out(v) i i! ways.Therefore, and the total number of decompositions of the acyclic digraph G is

On general walk and path decompositions
and the total flux as f (G) := The basic properties of walk decompositions such as existence criteria, simply rely on the usual Euler tours.We just add a new vertex so that for all v ∈ V we get flux(v) = 0.This also allows to compute (or bound) the total number of walk decompositions using the BEST theorem and the Matrix-tree theorem.
Decompositions into walks have the following matroid structure studied in [23].Sets of edges that are in distinct walks in a certain (minimal) decomposition are independent sets of a matroid of rank f (G) (adding with the number of components of G).This matroid is isomorphic to a cotransversal matroid.In fact, the sets of source (initial) edges of all f (G)-decompositions also form a collection of matroid bases.
For (simple) path decompositions there are many studies (e.g.[1,2,8,13,16,17,19]), most of which are around Gallai's conjecture.Note that f (G) is attainable minimal number of paths in a decomposition for acyclic digraphs; it was also shown in [1] that the minimal number of paths needed to decompose a transitive tournament digraph on n vertices is n 2 /4 .Gallai's conjecture states that every connected simple undirected graph with n vertices can be decomposed into at most n/2 paths; it is known that every such graph can be decomposed into at most n/2 paths or cycles [19].

G-Stirling functions
We use the following notation for multisets: the electronic journal of combinatorics 22(4) (2015), #P4.10 For a given digraph G, let i.e., M out is the set of all sub(multi)sets of V out .
Note that if for a k-decomposition, we have the sources I, then the corresponding sinks J = V in I − V out are determined uniquely.(Further, for any sources I we will just write sinks as where S i (k) the number of matchings in M(i) of size k.
Proof.Since S i (k) the number of matchings in M(i) of size k, the vertex i is unmatched out(i) − k times.Thus, from the bijection of Proposition 2.1, the vertex i is used exactly out(i) − i times as a source.Therefore, we get the formula by considering this argument for every vertex i = 1, . . ., n and using Proposition 2.1.

Theorem 3.2 ([15]
).The G-Stirling function S G satisfies the following properties: (ii) if S G (I) > 0 for some I ⊂ V out , then for any I , such that V out ⊇ I ⊃ I, we have S G (I ) > 0; (iii) Suppose that digraph G is built up from blocks B 1 , . . ., B m so that the indices of edges increase with respect to the order of blocks.Let e = (i, j) ∈ B m , G = G − e, I = I − {i}.Let k i be the number of repetitions of i in (J − {j}) {i} and r e be the number of edges in B m − e that end by i.Then the following recurrence relation holds for S G (I).Note that S G is different from Stirling (and Bell) numbers for graphs studied in [14], which count partitions of graph vertex set into independent sets.Although, for n = 1 (and several blocks) there is a correspondence between these definitions as noted in section 6.
In fact, M out (G) can be considered as a poset ordered by inclusion.Let us consider a subdomain of M out (G) at which S G takes positive values; define the poset whose elements (multisets) are ordered by inclusion.
Proposition 3.6.Let m(v) = |M (v)| be the size of a maximal matching.Then (i) P G has a unique maximal element V out , unique minimal element V 0 , where Here by m we denote the chain poset of m elements and m × s is the poset of ms elements defined as (cartesian) product of posets m, s (if s is empty, then put m×s = m).
(See e.g.[29]) Proof.First, from (i), (ii) of Theorem 3.2, V out is a unique maximal element and if V 0 is some minimal element, then for all V 0 ⊆ I ⊆ V out .Let us prove that is a unique minimal element.From Theorem 2.1, there is a principal decomposition with sources V 0 , so V 0 ∈ P G .If there is another minimal element V , then i out(i)−m(i) ∈ V for some i.This means that the vertex i has a matching of size greater than m(i), which is impossible.So, both items (i), (ii) clearly imply from these arguments.Corollary 3.7.If G is a cycle graph with edge labels given by The total number of principal decompositions |PD(G)| serves as a generalized Bell number.Let B G = |PD(G)|, then we have (5) If we now define an extension B G (J) as then applying the Möbius inversion formula on the poset P G we get and in particular Note that P G is isomorphic to a divisibility poset, and µ(I, J) and the kind of recurrence

Normal ordering in the Weyl algebra
Let K be a field of characteristic 0. The n-th Weyl algebra A n is an associative algebra over K defined by 2n generators x 1 , . . ., x n , ∂ 1 , . . ., ∂ n and relations where [a, b] = ab − ba is the commutator and δ i,j is the Kronecker symbol.The typical example of A n is the polynomial algebra with ∂ i considered as partial derivations d/dx i .The elements of types We will also write monomials in the equivalent form x i 1 . . .x is ∂ j 1 . . .∂ jp .All monomials x α ∂ β form a linear vector space basis of A n .When the element w of A n is expressed as a linear combination the electronic journal of combinatorics 22(4) (2015), #P4. 10 we say that w is normally ordered.Define the following subspaces of A n : n is the subalgebra of A n .We show that combinatorial meaning of coefficients in the normal ordering can be interpreted in terms of graph decompositions.Furthermore, we will consider the monomials of subspace A (0) n (otherwise, we may add fictive elements as shown in subsection 6.1).We associate every monomial w = x i 1 . . .
n be monomials.Then we have where G = ([n], E) with E = (block(w 1 ), . . ., block(w m )) (i.e. the indices of edges increase with respect to the order of blocks) and E) with E = (e 1 , . . ., e m ), we have where e = (i , j ) and the sum runs over all (multi)sets of sources I.
Corollary 4.3.If n = 1, then (8) gives the well-known formula where S(m, i) is Stirling number of the second kind.
Analogous to formula (8), the operators x i ∂ j + x j ∂ i give a decomposition formula for undirected graphs.Namely, we obtain the following result.
decompositions into increasing walks with sources I i∈I the electronic journal of combinatorics 22(4) (2015), #P4.10 The normal ordering decomposition formulas like (8), ( 9) are in particular useful when we sum through all decompositions of G or symmetrize over all permutations of the edge set.For w 1 , . . ., w m ∈ A n , let Then for undirected graph G we have Consider a case of computing the sum s + m .
Taking all permutations σ ∈ S m means that we permute the edges of G up to σ.Consider decompositions into increasing walks in that case.Suppose that we break the paths at vertices 2 Then the number of permutations σ ∈ S m for which the fragments 1 where B k is the Bell number, the number of partitions of set [k].
Proof.First we examine the approach using composition of differential operators.Note that A(m, n) corresponds to the number B G = I S G (I) (see def. ( 5)) of all principal decompositions of G = C m,n .Therefore, we can use the normal ordering expansion of differential operators to calculate A(m, n) = B G .If m = nk, then the composition of operators is In fact, A(m, n) is the sum of coefficients in the normal ordering expansion of the last expression.The sum of coefficients in expansion ( and the sum of coefficients in the normal ordering of (∂ It remains to use the well-known recurrence for Bell numbers So, the sum of coefficients A(nk, n) in the normal ordering of ( Similarly, if m = nk + r we have the composition and by the same argument it follows that the electronic journal of combinatorics 22(4) (2015), #P4.10 Bijective proof.Using Corollary 2.2 of Theorem 2.1 we look at matchings of edges at every vertex.Suppose m = nk and consider the vertex i, where 2 i n.We have Out(i) = {e i , e n+i , e 2n+i , . . ., e (k−1)n+i }.
We will prove that |M(i)| = B k+1 by establishing a bijection between matchings in M(i) and partitions of set [k + 1].Let M ∈ M(i) be any matching between In(i), Out(i).Construct set partition of [k + 1] as follows: (1) if (e an+i−1 , e bn+i ) ∈ M , where a b, then put a + 1, b + 2 in the same block; (2) the remaining elements of [k + 1] (that were not considered yet), put in separate blocks.
One can see that this properly defines the bijection.By applying a similar argument one can show that |M(1)| = B k and therefore, by Theorem 2.1, A(nk, n) = B k (B k+1 ) n−1 .The formula for A(nk + r, n) implies analogously.

Parity alternating partitions
For n = 2 we have the graph model with 2 vertices and n edges {e 1 , . . ., e n } such that all odd-indexed e 2i−1 are of type (1, 2) and all even-indexed e 2i are of type (2,1).
Denote by a(m) = A(m, 2) the total number of parity alternating partitions of [m].Then the following formulas hold Remark 5.2.The latter formulas mean that the number of parity alternating partitions of [m] is equal to the number of partitions of [m + 1] where elements in each block have the same parity.(The second sequence appears in OEIS, A124419 [28]).Bijectively, this fact can be described as follows: x, y are successive elements in the block of parity alternating partitions iff x, y + 1 are successive elements of the second type of partitions.Note that such reduction algorithm was applied to regular (noncrossing) set partitions in [6].
The values of a(m) can also be computed by the number of blocks.Let a(m, k) be the number of parity alternating partitions into k blocks and a(m, k, i) the number of parity alternating partitions into k blocks i of which have even maximal elements.We have a(m, k) = k i=0 a(m, k, i), a(m) = n k=1 a(m, k).
Proof.Consider parity alternating partitions of set [2m + 1] into k blocks, i of whose maximal elements are even.The element 2m + 1 can form a separate block contributing a(2m, k − 1, i) ways.Otherwise, 2m + 1 can be placed into blocks of parity alternating partitions of [2m] having k blocks and (i + 1) maximal even elements (since 2m + 1 will change the parity of one maximal element); this gives (i + 1)a(2m, k, i + 1) ways.

Combinatorial interpretations of coefficients in the normal ordering expansion
the electronic journal of combinatorics 22(4) (2015), #P4.10 can be extracted from Theorem 4.1 as follows.The elements x r i ∂ s i are not from A (0) 1 = x i ∂ i : i = 0, 1, . . . in general (i.e. if r i = s i ).To deal with this situation, we add fictive |r i − s i | new variables x i+1 or ∂ i+1 so that the monomial will belong to A (0) 1 , and the graph scheme can be applied.For example, x 2 ∂ 5 is transformed to x 3 2 x 2 ∂ 5 .Note that the new variables commute with all other and so we can freely move them in the normal ordering expansion.Using these new monomials, we construct the graph G according to the rules above.Thus, combinatorial meaning of S G (i) can be described as the number of principal decompositions of G having i sources at vertex 1.This interpretation is similar to the graph models studied in [3] (the model there is acyclic which is different to ours since for n = 1 we have loops).Remark 6.1.In fact, the normal ordering in the n-th Weyl algebra can be computed using the n = 1 case.For instance, we can restructure compositions as follows This view also helps to refine all possible multisets of sources and sinks I, J, since what coefficients are nonzero in every composition like (10) can be found.

The λ-Stirling numbers
Consider the graph with n = 1 vertex and suppose it is built up from blocks of loops (1, 1) of λ 1 , λ 2 , . . .edge sizes.Principal decompositions on this model require that the edges within one block cannot lie on the same walk.This setting clearly corresponds to partitions of the set [m] (m is a total number of edges), where the first λ 1 elements are in distinct subsets, the next λ 2 elements are also in distinct subsets, and so on.The coefficients S G (I) present a generalization of Stirling numbers of the second kind on such restricted partitions.In this section we study these generalized Stirling numbers.We also introduce a generalization of Stirling numbers of the first kind, which can be considered as dual to the second.We call these numbers the λ-Stirling numbers.
Fix the sequence of nonnegative integers, and let So λ is a kind of 'infinite' integer partition, q is the analog of quotient and r is the analog of remainder.
Definition.The λ-Stirling numbers of second and first kinds S λ (n, k), C λ (n, k) are defined as follows: the electronic journal of combinatorics 22(4) (2015), #P4.10 • S λ (n, k) is the number of partitions of [n] into k blocks such that the first λ 1 elements of [n] are in distinct blocks, the next λ 2 elements are in distinct blocks, and so on; the remaining r n elements are also in distinct blocks.
• C λ (n, k) is the number of permutations of [n] with k cycles such that non-minimal elements of the first λ 1 cycles are greater than all minimal elements of these λ 1 cycles; non-minimal elements of the next λ 2 cycles are greater than all minimal elements of these λ 2 cycles, and so on; and the remaining r k cycles are singletons (i.e.consist of one element).
Clearly if λ = (1, 1, . ..), then C λ (n, k), S λ (n, k) are just the usual Stirling numbers.The case λ = (r, 1, 1, . ..) corresponds to the r-Stirling numbers of first and second kinds introduced in [5].Generalized Stirling numbers of the second kind S r,s (n, k) that arise from the expansion have been studied in the bosons normal ordering problem [4,24] (see also [12]).Our definition gives a natural (and simple) combinatorial interpretation to the case S r,r (n, k) and also solves an inverse problem, where Stirling numbers of the first kind arise.Note that the general formulas for S r,s (n, k) given in [24] can be used to compute our numbers S G , B G (defined in Section 2).Interpretation to S r,r (n, k) with colorings of complete graphs introduced in [7] is very close to the meaning of S λ (n, k) (λ = (r, r, . ..)), since any r elements that cannot be in the same subset can be viewed as proper colorings of a component complete graph K r ; on the other hand, this definition corresponds to Stirling numbers for graphs in the sense of counting partitions of a vertex set into independent sets (see [14]).The numbers S λ (n, k) were introduced in [25] as (r 1 , . . ., r p )-Stirling numbers of the second kind.In order to be consistent with the corresponding Stirling numbers of the first kind C λ (n, k), we define these numbers over a general sequence λ relating it with integer partitions, since it somehow acts (as integer partition) on first (or resp.second) argument of these Stirling numbers.In fact, by polynomial relations (18), (19) shown below, the λ-Stirling numbers correspond to a case of the multiparameter noncentral Stirling numbers introduced in [11].This also leads to a case of a general study of connection constants between persistent sequences of polynomials [10].
Then the number of ways to distribute the elements of n into k ordered nonempty sets is equal to This easily implies from the combinatorial interpretation of S λ (n, k).If blocks are ordered, then we multiply the number of ways by k!.The elements λ 1 +• • •+λ j−1 +1, . . ., λ 1 +• • •+λ j (which all in distinct blocks) can be changed to the repetition j λ j ; that was calculated λ j !times.
the electronic journal of combinatorics 22(4) (2015), #P4.10 The last formula gives Let D λ n be the operator defined as D λ n := D λ 0 • • • D λq D r .Similar to equation (12) we can obtain that Theorem 6.3.The numbers S λ (n, k), C λ (n, k) have the following properties.(i) Recurrence relations with with (ii) Expansions with differential operators (iii) Polynomial expansions (vi) The general formula for S λ (n, k) is given by (vii) The following recurrence relations hold Proof.(i) Recurrence relations.We can show that the number of described partitions has the same recurrence as (14).Note that the number of ways is 0 when k < λ 1 or k > n.If we consider the element n, then two cases are possible.If n forms a separate block, then we have the number of ways to partition [n − 1] into k − 1 parts over partition λ.If n if placed in the block with some other elements except the restricted; this can be done in (k − r n−1 ) ways of any of partitions of [n − 1] into k blocks over partitions λ.This argument implies the needed recurrence for S λ (n, k).
We show that the described number of ways satisfies the same recurrence as (15).Note that the number of ways is 0 when k < λ 1 or k > n.Consider the element n and two cases.If n form a singleton separate cycle, then the number of corresponding ways is the number of permutations of [n − 1] having k − 1 cycles with the properties for λ partition of k − 1.If n is in cycle with the other elements, then we can put n in cycles after any element except last r k singletons.This gives (n − 1 − r k ) ways for any permutation of [n − 1] with k cycles and the described partition property.This argument clearly implies the needed recurrence for C λ (n, k).(iii) Polynomial expansions.
Applying the derivation operation to the function x t with a real parameter t, expansions ( 16), (17) The last two identities are polynomial relations in t and hold for all t which imply (18), (19).
(vi) The general formula for S λ (n, k).We will show that this formula holds using combinatorial interpretation of S λ (n, k) and the inclusion-exclusion principle.Suppose that blocks are ordered.Let us enumerate them as 1, . . ., k.
Denote by A i (1 i k) the set of corresponding restricted (up to λ) arrangements of [n] into k ordered blocks such that the i-th block is empty.Let A be the number of all restricted arrangements of [n] into k blocks (some of them might be empty).Then it is clear that k!S λ (n, k)  (vii) Recurrence relations.Recurrence (25).Suppose that j elements of the last r n in [n] are singleton blocks.We can choose these elements in rn j ways.The remaining (r n − j) elements should be put in distinct (k − j) blocks of any of S λ (n − r n , k − j) partitions, which can be done (k − j) rn−j times.
Recurrence (26).We may choose the needed r k singleton cycles from the last j elements of [n].This can be done in j r k ways.The remaining (j − r k ) elements should be put in the first (n − j) cycles of any of C λ (n − j, k − r k ) permutations, which can be done (n − r k − 1) j−r k times.

3 Figure 1 :
Figure 1: A digraph with an ordered set edges.
as follows S G (I) := the number of principal decompositions of G with sources I.If n = 1, then S G (I) corresponds to Stirling number of the second kind S(m, k) where |I| = k and digraph G has m labeled loops (1, 1).Proposition 3.1.

Table 1 :
Small values of a(m) and a(m, k).Proposition 5.3.Recurrence relations for a(m, k, i) are given by