# integer partition

## integer partition

[′int·ə·jər pär′tish·ən]
(mathematics)
For a positive integer n, a nonincreasing sequence of positive integers whose sum equals n.
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and NEGPAR([pi]) to be the integer partition with part [absolute value of [[pi].sub.i]] for each [[pi].sub.i] < 0.
If [lambda] = ([[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.r]) is the multi-degree of w then {w} is just a [lambda]-tabloid, where [lambda] may be, without loss of generality, assumed an integer partition of n.
"Each step in the story is a work of art,' Dyson says, "and the story as a whole is a sequence of episodes of rare beauty, a drama built out of nothing but numbers and imagination." INTEGER PARTITION NUMBER 1 1 2 2 3 3 4 5 5 7 6 11 7 15 8 22 9 30 10 42 11 56 12 77 13 101 14 135 15 176 16 231 17 297 18 385 19 490 20 627 21 792 22 1,002 PARTITIONS OF 4 RANK GROUP NUMBER 4 4 - 1 = 3 3 3 + 1 3 - 2 = 1 1 3 + 2 2 - 2 = 0 0 2 + 1 + 1 2 - 3 = -1 4 1 + 1 + 1 + 1 1 - 4 = -3 2
We call signed Young diagrams (briefly SYD) an ordered couple D = [D.sub.1] : [D.sub.2], where [D.sub.1] is a Young diagram of an integer partition built with decreasing columns instead with decreasing rows, and [D.sub.2] is a Young diagram of an integer partition built with increasing columns.
[i.sub.l] there is an integer partition [lambda](w) = ([[lambda].sub.1], [[lambda].sub.2], ...) whose ith part [[lambda].sub.i] equals the number occ(i, l) of occurrences of i in w.
Denote by [LAMBDA]([pi]) the unique integer partition defined by (#[[pi].sub.1], ..., #[[pi].sub.k]) if [pi] = ..., nk} with #[[pi].sub.1] [greater than or equal to] #[[pi].sub.2] [greater than or equal to] ...
The proof of Theorem 1 requires working knowledge of the theory of integer partition, especially its representation as Young diagrams, the Hasse diagram of the Young lattice, and the RSK-algorithm for filling positive integers to obtain the beginning of some standard Young tableau.
Let [lambda] = ([[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.p]) [??] n an integer partition of n with l([lambda]) = p parts sorted in decreasing order.
For each integer partition [lambda] = ([[lambda].sub.1], [[lambda].sub.2], ..., [[lambda].sub.l]), the monomial symmetric function corresponding to [lambda] is the sum
The computation of the HP-series of k[[y.sub.1],[y.sub.2], ...) allows an easy combinatorial interpretation: the weight of a monomial [y.sup.[alpha]] can be interpreted, as we have seen already in Section 3, as an integer partition. By factoring out L(I) we factor out all monomials [y.sup.[alpha]] which contain as factors an [y.sup.2.sub.i] or [y.sub.i][y.sub.i+1].
An integer partition [lambda] of a non-negative integer n is a multiset of positive integers whose sum is n.
It is common to associate to a set-partition the integer partition formed by the non-increasing sequence of the part sizes.

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