New partition identities for odd w odd

In this note we conjecture Rogers-Ramanujan type colored partition identities for an array with odd number of rows w such that the first and the last row consist of even positive integers. In a strange way this is different from the partition identities for the array with odd number of rows w such that the first and the last row consist of odd positive integers -- the partition identities conjectured by S. Capparelli, A. Meurman, A. Primc and the author and related to standard representations of the affine Lie algebra of type $C^{(1)}_\ell$ for $w=2\ell+1$. The conjecture is based on numerical evidence.


Introduction
We write a partition of positive integer n in terms of frequencies f j -the number of occurrences of the part j in the partition (1.1) n = j∈N It is clear that f j = 0 for all but finitely many j ∈ N and that the partition (1.1) is determined by its sequence of frequencies (f i | i ∈ N).
The partition identities of Rogers (1894), Ramanujan (1913) and Schur (1917) for k = 1, and the partition identities of Gordon (1961) for k ≥ 2, can be stated as: Let 0 ≤ a ≤ k.The number of partitions of n such that (1.2) f j + f j+1 ≤ k for all j and (1.3) f 1 ≤ a equals the number of partitions of n into parts ≡ 0, ±(a + 1) mod (2k + 3).
The conditions (1.2) on frequencies of two adjacent numbers are called the difference conditions, and the condition (1.3) on the frequency of number 1 is called the initial condition.There are some other similar partition identities stating that the number of partitions of n satisfying certain difference & initial conditions is equal to the number of partitions of n with parts satisfying certain congruence conditions; these identities are often called the classical Rogers-Ramanujan type identities-see [A].
On the other side, some parts of representation theory of affine Kac-Moody Lie algebras lead to Rogers-Ramanujan type colored partition identities.
Let N 1 , . . ., N r , r ≥ 2, be non-empty subsets of the set of positive integers N and let N be the multiset (1.4) If a positive integer a appears in several subsets N i , then a appears in the multiset N several times.To see these elements in N as different, for each positive integer a we may "color" a ∈ N j with a "color" j by writting a j = (a, j) ∈ N j × {j}, and then write (1.4) in terms of sets as We say that elements in the multiset N appear in r colors.In this note a colored partition of positive integer n on the multiset N is It is clear that f a = 0 for all but finitely many a ∈ N and that the partition (1.6) is determined by its "sequence" of frequencies (f a | a ∈ N ).
Example 1.1. where Then parts a of colored partitions (1.6) for N = N 1 ∪ N 2 appear in two colors, 1 and 2: parts ≡ 2, 8 mod 10 appear in both colors, and parts ≡ 1, 4, 5, 6, 9 mod 10 appear only in color 2. Note that the generating function for colored partitions (1.6) is the infinite periodic product with modulus 10: (1.7) j≡1,2,2,4,5,6,8,8,9 mod 10 Lepowsky and Wilson gave in [LW] a Lie theoretic interpretation of the classical Rogers-Ramanujan type partition identities in terms of characters of standard modules L A (1) 1 (Λ) for affine Kac-Moody Lie algebra of the type A (1) 1 .After their discovery it was expected that for each standard module L g(A) (Λ) for any affine Lie algebra g(A) (cf.[K]) there is a Rogers-Ramanujan type partition identity, where 1 is just "the smallest one" on the list of all affine Lie algebras: 4 .However, besides several sporadic results beyond 1 , so far this goal is not achieved.In [CMPP] Rogers-Ramanujan type partition identities are conjectured for all standard C (1) ℓ -modules, stating that the number of colored partitions of n with parts satisfying certain congruence conditions is equal to the number of colored partitions (1.6) for a multiset N = N 2ℓ+1 composed of ℓ copies of N and an additional copy of (2N + 1), satisfying difference & initial conditions similar to (1.2)-( 1.3) , but much more complicated.Moreover, in [CMPP] another series of similar partition identities is conjectured for a multiset N = N 2ℓ composed of ℓ copies of N, satisfying certain difference & initial conditions, but with no obvious connection to representation theory of affine Lie algebras.
In this note we conjecture yet another Rogers-Ramanujan type colored partition identities for a multiset N = N odd 2ℓ−1 , somewhat similar to the conjectured identities for standard C (1) ℓ -modules, but again with no obvious connection to representation theory of affine Lie algebras.

Arrays with odd width w and even first row
Let N = N odd 5 be the colored array of natural numbers with 5 rows (2.1) N is a multiset composed of 2 copies of N and an additional copy of 2N, but its elements are arranged in such a way that in the first row are even numbers and that numbers increase by one going to the right on any diagonal.We consider colored partitions where f a is the frequency of the part a ∈ N in the colored partition (2.2) of n.It is clear that f a = 0 for all but finitely many a ∈ N and that the colored partition (2.2) is determined by its array F of frequencies We say that two elements in the array F are adjacent if they are simultaneously on two adjacent rows and two adjacent diagonals.For example, f 51 and f 71 in the second row are adjacent to f 61 in the first row and, just as well, adjacent to f 62 in the third row.We say that the set 1 {a 1 , a 2 , a 3 , . . .} is a downward path Z in the array F if a i is in the i-th row and if (a i , a i+1 ) is a pair of two adjacent elements for all i.For example, Z = {f 61 , f 51 , f 62 , f 72 , f 63 } is a downward path in F and there are altogether 2 4 downward paths through f 61 in the first row.
Let k be a positive integer.We say that the frequency array F satisfies level k difference conditions if (2.4) m∈Z m ≤ k for all downward paths Z in F .
Note that the level k difference conditions for a frequency array F is similar to difference conditions (1.2) for a sequence of frequencies ( We say that an array of frequencies F is (k 0 , k 1 , k 2 , k 3 ) odd -admissible if the extended array of frequencies (2.5) 1 or the sequence satisfies the level k difference conditions, that is (2.6) m∈Z m ≤ k for all downward paths Z in F (k0,k1,k2,k3)odd .
Note the difference between (2.4) and (2.6): (k 0 , k 1 , k 2 , k 3 ) odd -admissible frequency array F satisfies the level k difference conditions (2.4), but in addition to that there are new conditions on the frequencies at the beginning of the array, somewhat similar to initial condition (1.3), but much more complicated.For example, f 11 in the second row must be ≤ k 2 because of (2.6) for the downward path Z = {k 3 , f 11 , 0, k 1 , k 0 }.We say that colored partitions (2.2) with (k 0 , k 1 , k 2 , k 3 ) odd -admissible arrays of frequencies (2.3) are (k 0 , k 1 , k 2 , k 3 ) odd -admissible colored partitions.
2 by using a slightly modified code 21AAIC in [CMPP] with built in option to choose even numbers in the top row (for p=0) or to choose odd numbers in the top row (for p=1);