Using Rota's Umbral Calculus to Enumerate Stanley's P-Partitions

By Shalosh B. Ekhad and Doron Zeilberger

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[Appeared in Advances in Applied Mathematics v. 41 (2008), 206-217.]
Dedicated to Two Combinatorial Giants:
Gian-Carlo Rota   ז צ " ל   (April 27, 1932-Apri 18, 1999)
and
Richard Stanley   ש ל י" ט א   (b. June 23, 1944)
Written: June 12, 2007.

I didn't contribute anything for Gian-Carlo Rota's 64th birthday collection or Richard Stanley's 60th birthday collection, since at the time I didn't have anything relevant, and I don't like to submit a paper that is not closely related to the research interests of the honorees. So this paper is my belated present to these two heroes of mine, and my coauthor, Shalosh B. Ekhad, really enjoyed computing these lovely brainchildren of Richard Stanley using Gian-Carlo Rota's powerful method of the Umbral Calculus.
Important: This article is accompanied by Maple package RotaStanley that computes generating functions enumerating P-partitions (in the sense of Stanley) directly, and more impressively, indirectly, using Rota's Umbral Calculus as it is described in this article. Even more impressively, it finds computer-generated functional recurrences for generating functions of P-partitions for families of posets that are obtained by repeated "grafting".

Sample Input and Output For Generating Functions of P-partitions for Specific Posets

• In order to compute the generating functions (in q) for the Boolean Lattices of dimensions n=1,2,3, the input would yield the output

• In order to compute the generating function for P-partitions for the lattice of partitions into distinct parts with largest part less than equal to n, for n=1, 2,3,4, the input would yield the output

• In order to compute the generating function for P-partitions for the Gaussian Lattice G(a,b) (the lattice of partitions into at most b parts, with each part at most a) for a,b between 2 and 3, the input would yield the output

• In order to compute the generating functions, and first few terms, for solid partitions who live in a 2 by 2 by n by infinity box with n between 2 and 5, the input would yield the output

• In order to compute the generating functions, and first few terms, for stair-case shapes (going in either direction), the input would yield the output

Sample Input and Output For Functional-Recurrences for Generating Functions of P-partitions for Powers of a Given Poset

Below are a few examples of computer-generated functional-recurrence equations for powers of given posets. Note that these are fully general theorems. We use procedure UmbralRecurrenceSingle of the package RotaStanley.
• In order to get the functional-recurrence equation for powers of the two-dimensional Boolean-lattice (alias diamond) where the last vertex of the previous link is identified with the first one of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of the two-dimensional Boolean-lattice (alias diamond) where the last two vertices of the previous link are identified with the first two of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of the poset that is a 2-rowed tableau each row with 3 cells, and the bottom row is shifted one cell to the left, where the last three vertices of the previous link are identified with the first three of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of the poset that is a 2-rowed tableau each row with 3 cells, and no shift, where the last three vertices of the previous link are identified with the first three of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of the poset that is a 2-rowed tableau each row with 3 cells, and the bottom row is shifted one cell to the right, where the last three vertices of the previous link are identified with the first three of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of a certain hexagon where the last vertex of the previous link is identified with the first vertex of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of a certain hexagon where the last two vertices of the previous link are identified with the first two vertices of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of another hexagon where the last vertex of the previous link is identified with the first vertex of the next, the input would yield the output

• In order to get the functional-recurrence equation for powers of that other hexagon where the last two vertices of the previous link are identified with the first two vertices of the next, the input would yield the output

Doron Zeilberger's List of Papers