Trac #34373: Implement multimajor index for permutations

We add an implementation for the multimajor index
of a permutation with respect to a combination of
the same size, as described in:

> Armin Jöllenbeck and Manfred Schocker. Cyclic
> characters of symmetric groups. J. Algebraic Combin.,
> 12 (2000), 155-161. [https://doi.org/10.1023/A:1026592027019
doi:10.1023/A:1026592027019]

URL: https://trac.sagemath.org/34373
Reported by: gh-25shriya
Ticket author(s): Amrutha P, Shriya M, Divya Aggarwal, Aritra
Bhattacharya
Reviewer(s): Amrutha P, Aritra Bhattacharya, Julian Rüth, Samuel
Lelièvre

src/doc/en/reference/references/index.rst

@@ -3330,6 +3330,11 @@ REFERENCES:
             *Combinatorics of symmetric designs*.
             Cambridge University Press, 2006.
 
+.. [JS2000] \A. Joellenbeck, M. Schocker.
+            "Cyclic Characters of Symmetric Groups".
+            J. Algebraic Combin., **12** (2000), 155-161.
+            :doi:`10.1023/A:1026592027019`.
+
 .. [JS2010] \B. Jones, A. Schilling.
             "Affine structures and a tableau model for `E_6` crystals",
             J. Algebra. **324** (2010). 2512-2542.

src/sage/combinat/permutation.py

@@ -221,12 +221,18 @@
   finite Weyl group to make it more uniform with :class:`SymmetricGroup`.
   Added ability to compute the conjugacy classes.
 
+- Amrutha P, Shriya M, Divya Aggarwal (2022-08-16): Added Multimajor Index.
+
 Classes and methods
 ===================
 """
 
 # ****************************************************************************
 #       Copyright (C) 2007 Mike Hansen <mhansen@gmail.com>
+#                     2022 Amrutha P <amruthap1916@gmail.com>
+#                     2022 Shriya M <25shriya@gmail.com>
+#                     2022 Divya Aggarwal <divyaa@iiitd.ac.in>
+#
 #
 # This program is free software: you can redistribute it and/or modify
 # it under the terms of the GNU General Public License as published by
@@ -3365,6 +3371,52 @@ def major_index(self, final_descent=False) -> Integer:
         descents = self.descents(final_descent)
         return sum(descents)
 
+    def multi_major_index(self, composition):
+        r"""
+        Return the multimajor index of this permutation with respect to ``composition``.
+
+        INPUT:
+
+        - ``composition`` -- a composition of the :meth:`size` of this permutation
+
+        EXAMPLES::
+
+            sage: p = Permutation([5, 6, 2, 1, 3, 7, 4])
+            sage: p.multi_major_index([3, 2, 2])
+            [2, 0, 1]
+            sage: p.multi_major_index([7]) == [p.major_index()]
+            True
+            sage: p.multi_major_index([1]*7)
+            [0, 0, 0, 0, 0, 0, 0]
+            sage: Permutation([]).multi_major_index([])
+            []
+
+        TESTS::
+
+            sage: p.multi_major_index([1, 3, 3, 7])
+            Traceback (most recent call last):
+            ...
+            ValueError: size of the composition should be equal to size of the permutation
+
+        REFERENCES:
+
+        - [JS2000]_
+        """
+        composition = Composition(composition)
+        if self.size() != composition.size():
+            raise ValueError("size of the composition should be equal to size of the permutation")
+        descents = self.descents()
+        partial_sum = [0] + composition.partial_sums()
+        multimajor_index = []
+        for j in range(1, len(partial_sum)):
+            a = partial_sum[j-1]
+            b = partial_sum[j]
+            from bisect import bisect_right, bisect_left
+            start = bisect_right(descents, a)
+            end = bisect_left(descents, b)
+            multimajor_index.append(sum(descents[start: end])-(end-start)*a)
+        return multimajor_index
+
     def imajor_index(self, final_descent=False) -> Integer:
         """
         Return the inverse major index of the permutation ``self``, which is