659 research outputs found
Card deals, lattice paths, abelian words and combinatorial identities
We give combinatorial interpretations of several related identities
associated with the names Barrucand, Strehl and Franel, including one for the
Apery numbers. The combinatorial constructs employed are derangement-type card
deals as introduced in a previous paper on Barrucand's identity, labeled
lattice paths and, following a comment of Jeffrey Shallit, abelian words over a
3-letter alphabet.Comment: 10 pages, LaTe
Jordan and Smith forms of Pascal-related matrices
We present matrix identities which yield respectively the Jordan canonical
form of the Pascal matrix P_n = (i -1 choose j -1)_{1 <= i,j <= n} modulo a
prime, the eigenvectors of (i choose j)_{1 <= i,j <= n}, and the Smith normal
form of powers of P_n - I_n.Comment: LaTeX, 4 page
On conjugates for set partitions and integer compositions
There is a familiar conjugate for integer partitions: transpose the Ferrers
diagram, and a conjugate for integer compositions: transpose a Ferrers-like
diagram. Here we propose a conjugate for set partitions and show that it
interchanges # singletons and # adjacencies. Its restriction to noncrossing
partitions cropped up in a 1972 paper of Kreweras. We also exhibit an analogous
pairs of statistics interchanged by the composition conjugate.Comment: Improved definition of conjugate partition; noncrossing partitions
considered. 7 page
A sign-reversing involution to count labeled lone-child-avoiding trees
We use a sign-reversing involution to show that trees on the vertex set [n],
considered to be rooted at 1, in which no vertex has exactly one child are
counted by 1/n sum_{k=1}^{n} (-1)^(n-k) {n}-choose-{k} (n-1)!/(k-1)! k^(k-1).
This result corrects a persistent misprint in the Encyclopedia of Integer
Sequences.Comment: 4 page
Counting stabilized-interval-free permutations
A stabilized-interval-free (SIF) permutation on [n]={1,2,...,n} is one that
does not stabilize any proper subinterval of [n]. By presenting a decomposition
of an arbitrary permutation into a list of SIF permutations, we show that the
generating function A(x) for SIF permutations satisfies the defining property:
[x^(n-1)] A(x)^n = n! . We also give an efficient recurrence for counting SIF
permutations.Comment: latex, 6 page
The number of bar{3}bar{1}542-avoiding permutations
We confirm a conjecture of Lara Pudwell and show that permutations of [n]
that avoid the barred pattern bar{3}bar{1}542 are counted by OEIS sequence
A047970. In fact, we show bijectively that the number of bar{3}bar{1}542
avoiders of length n with j+k left-to-right maxima, of which j initiate a
descent in the permutation and k do not, is {n}-choose-{k} j!
StirlingPartition{n-j-k}{j}, where StirlingPartition{n}{j} is the Stirling
partition number.Comment: 5 pages, 2 figures, minor correction
Kreweras's Narayana number identity has a simple Dyck path interpretation
We show that an identity of Kreweras for the Narayana numbers counts Dyck
paths with a given number of peaks by number of peak plateaus, where a peak
plateau is a run of consecutive peaks that is immediately preceded by an upstep
and followed by a downstep.Comment: 2 page
The 136th manifestation of C_n
We show bijectively that the Catalan number C_n counts Dyck (n+1)-paths in
which the terminal descent is of even length and all other descents to ground
level (if any) are of odd length.Comment: 3 pages, LaTe
An identity for the central binomial coefficient
We find the joint distribution of three simple statistics on lattice paths of
n upsteps and n downsteps leading to a triple sum identity for the central
binomial coefficient {2n}-choose-{n}. We explain why one of the constituent
double sums counts the irreducible pairs of compositions considered by Bender
et al., and we evaluate some of the other sums.Comment: 5 page
Flexagons yield a curious Catalan number identity
Hexaflexagons were popularized by the late Martin Gardner in his first
Scientific American column in 1956. Oakley and Wisner showed that they can be
represented abstractly by certain recursively defined permutations called pats,
and deduced that they are counted by the Catalan numbers. Counting pats by
number of descents yields the curious identity
Sum[1/(2n-2k+1)binom{2n-2k+1}{k}binom{2k}{n-k},k=0..n] = C(n), where only the
middle third of the summands are nonzero.Comment: 4 pages, shorter method, suggested by Ira Gessel; other (minor)
improvement
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