Geodesic
RSK tableaux and the weak order on fully commutative permutations
Emily Gunawan1 ; Jianping Pan2 ; Heather M. Russell3 ; Bridget Tenner4
1University of Oklahoma
2North Carolina State University
3University of Richmond
4DePaul University
The electronic journal of combinatorics, Tome 30 (2023) no. 4
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For each fully commutative permutation, we construct a “boolean core,” which is the maximal boolean permutation in its principal order ideal under the right weak order. We partition the set of fully commutative permutations into the recently defined crowded and uncrowded elements, distinguished by whether or not their RSK insertion tableaux satisfy a sparsity condition. We show that a fully commutative element is uncrowded exactly when it shares the RSK insertion tableau with its boolean core. We present the dynamics of the right weak order on fully commutative permutations, with particular interest in when they change from uncrowded to crowded. In particular, we use consecutive permutation patterns and descents to characterize the minimal crowded elements under the right weak order.
DOI : 10.37236/11877
Classification : 05E10, 05A05, 06A07, 20F55
Mots-clés : Boolean permutation, fully commutative permutation, permutation pattern, Robinson-Schensted-Knuth correspondence, reduced word, weak order

Emily Gunawan  1   ; Jianping Pan  2   ; Heather M. Russell  3   ; Bridget Tenner  4

1 University of Oklahoma
2 North Carolina State University
3 University of Richmond
4 DePaul University 
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     title = {RSK tableaux and the weak order on fully commutative permutations},
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Emily Gunawan; Jianping Pan; Heather M. Russell; Bridget Tenner. RSK tableaux and the weak order on fully commutative permutations. The electronic journal of combinatorics, Tome 30 (2023) no. 4. doi: 10.37236/11877

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